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import os
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib
import numpy as np

import rpy2.robjects as robjects
from rpy2.robjects import pandas2ri, vectors, numpy2ri

from rich.console import Console
from rich.table import Table

import DataExtractor
import datetime

import statsmodels.api as sm
from statsmodels.tsa.arima.model import ARIMA
from statsmodels.graphics.tsaplots import plot_pacf, plot_acf
from statsmodels.tsa.stattools import adfuller, kpss
from statsmodels.stats.diagnostic import het_breuschpagan, het_white, acorr_ljungbox

from scipy.stats import kurtosis, shapiro, kstest, norm, nct, genlogistic
from scipy.stats import fit
from scipy.optimize import differential_evolution

import warnings
warnings.filterwarnings('ignore')

from arch import arch_model

from scipy.stats import ttest_1samp
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NP_RANDOM_SEED = 1234567890
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r : robjects.R = robjects.r
r['source']('./R/R_Imports.R')

pandas2ri.activate()

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* installing *source* package ‘strucchange’ ...
** package ‘strucchange’ successfully unpacked and MD5 sums checked
** using staged installation
** libs
using C compiler: ‘x86_64-conda-linux-gnu-cc (Anaconda gcc) 11.2.0’

x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c strucchange_functions.c -o strucchange_functions.o
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x86_64-conda-linux-gnu-cc -shared -L/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/lib -Wl,-O2 -Wl,--sort-common -Wl,--as-needed -Wl,-z,relro -Wl,-z,now -Wl,--disable-new-dtags -Wl,--gc-sections -Wl,-rpath,/home/edoman000/miniconda3/envs/mpsmf_env/lib -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib -L/home/edoman000/miniconda3/envs/mpsmf_env/lib -o strucchange.so strucchange_functions.o strucchange_init.o -llapack -lblas -lgfortran -lm -lgomp -lquadmath -lpthread -L/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/lib -lR

installing to /home/edoman000/miniconda3/envs/mpsmf_env/lib/R/library/00LOCK-strucchange/00new/strucchange/libs
** R
** data
*** moving datasets to lazyload DB
** demo
** inst
** byte-compile and prepare package for lazy loading
** help
*** installing help indices
** building package indices
** installing vignettes
** testing if installed package can be loaded from temporary location
** checking absolute paths in shared objects and dynamic libraries
** testing if installed package can be loaded from final location
** testing if installed package keeps a record of temporary installation path
* DONE (strucchange)
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* installing *source* package ‘SupMZ’ ...
** package ‘SupMZ’ successfully unpacked and MD5 sums checked
** using staged installation
** R
** data
*** moving datasets to lazyload DB
** byte-compile and prepare package for lazy loading
** help
*** installing help indices
** building package indices
** testing if installed package can be loaded from temporary location
** testing if installed package can be loaded from final location
** testing if installed package keeps a record of temporary installation path
* DONE (SupMZ)
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R[write to console]: downloaded 161 KB


* installing *source* package ‘tseries’ ...
** package ‘tseries’ successfully unpacked and MD5 sums checked
** using staged installation
** libs
using C compiler: ‘x86_64-conda-linux-gnu-cc (Anaconda gcc) 11.2.0’
using Fortran compiler: ‘GNU Fortran (Anaconda gcc) 11.2.0’

x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c arma.c -o arma.o
x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c bdstest.c -o bdstest.o
x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c boot.c -o boot.o
x86_64-conda-linux-gnu-gfortran  -fpic  -fopenmp -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c  cfuncs.f90 -o cfuncs.o
x86_64-conda-linux-gnu-gfortran  -fpic  -fopenmp -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c dsumsl.f -o dsumsl.o

dsumsl.f:1004:72:

 1004 |  10      W(I) = A*X(I) + Y(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 10 at (1)
dsumsl.f:1123:72:

 1123 |  10      Y(I) = S
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 10 at (1)
dsumsl.f:1137:72:

 1137 |  10      X(I) = Y(I) / Z(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 10 at (1)
dsumsl.f:1141:72:

 1141 |  30      X(I) = Y(I) * Z(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 30 at (1)
dsumsl.f:1224:72:

 1224 |  30      Z(I) = CY * Z(I)  -  CS * W(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 30 at (1)
dsumsl.f:1908:72:

 1908 |  10      STEP(I) = G(I) / GNORM
      |                                                                        1
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dsumsl.f:1923:72:

 1923 |  20           STEP(I) = -NWTSTP(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 20 at (1)
dsumsl.f:1941:72:

 1941 |  40           STEP(I) = T * NWTSTP(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 40 at (1)
dsumsl.f:1955:72:

 1955 |  60           STEP(I) = T * DIG(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 60 at (1)
dsumsl.f:1982:72:

 1982 |  80      STEP(I) = T1*DIG(I) + T2*NWTSTP(I)
      |                                                                        1
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dsumsl.f:2226:72:

 2226 |  10      X(I) = Y(I)
      |                                                                        1
Warning: Fortran 2018 deleted feature: DO termination statement which is not END DO or CONTINUE with label 10 at (1)

x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c formats.c -o formats.o
x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c garch.c -o garch.o
x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c init.c -o init.o
x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c ppsum.c -o ppsum.o
x86_64-conda-linux-gnu-cc -I"/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/include" -DNDEBUG   -DNDEBUG -D_FORTIFY_SOURCE=2 -O2 -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -I/home/edoman000/miniconda3/envs/mpsmf_env/include -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib    -fpic  -march=nocona -mtune=haswell -ftree-vectorize -fPIC -fstack-protector-strong -fno-plt -O2 -ffunction-sections -pipe -isystem /home/edoman000/miniconda3/envs/mpsmf_env/include -fdebug-prefix-map=/workspace/croot/r-base_1695428141831/work=/usr/local/src/conda/r-base-4.3.1 -fdebug-prefix-map=/home/edoman000/miniconda3/envs/mpsmf_env=/usr/local/src/conda-prefix  -c tsutils.c -o tsutils.o
x86_64-conda-linux-gnu-cc -shared -L/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/lib -Wl,-O2 -Wl,--sort-common -Wl,--as-needed -Wl,-z,relro -Wl,-z,now -Wl,--disable-new-dtags -Wl,--gc-sections -Wl,-rpath,/home/edoman000/miniconda3/envs/mpsmf_env/lib -Wl,-rpath-link,/home/edoman000/miniconda3/envs/mpsmf_env/lib -L/home/edoman000/miniconda3/envs/mpsmf_env/lib -o tseries.so arma.o bdstest.o boot.o cfuncs.o dsumsl.o formats.o garch.o init.o ppsum.o tsutils.o -lblas -lgfortran -lm -lgomp -lquadmath -lpthread -lgfortran -lm -lgomp -lquadmath -lpthread -L/home/edoman000/miniconda3/envs/mpsmf_env/lib/R/lib -lR

installing to /home/edoman000/miniconda3/envs/mpsmf_env/lib/R/library/00LOCK-tseries/00new/tseries/libs
** R
** data
** byte-compile and prepare package for lazy loading
** help
*** installing help indices
** building package indices
** testing if installed package can be loaded from temporary location
** checking absolute paths in shared objects and dynamic libraries
** testing if installed package can be loaded from final location
** testing if installed package keeps a record of temporary installation path
* DONE (tseries)
R[write to console]: 

R[write to console]: 
R[write to console]: The downloaded source packages are in
	‘/tmp/RtmpKuhfBE/downloaded_packages’
R[write to console]: 
R[write to console]: 

R[write to console]: Updating HTML index of packages in '.Library'

R[write to console]: Making 'packages.html' ...
R[write to console]:  done

R[write to console]: Loading required package: zoo

R[write to console]: 
Attaching package: ‘zoo’


R[write to console]: The following objects are masked from ‘package:base’:

    as.Date, as.Date.numeric


R[write to console]: Loading required package: sandwich

R[write to console]: Registered S3 method overwritten by 'quantmod':
  method            from
  as.zoo.data.frame zoo 

R[write to console]: 
    ‘tseries’ version: 0.10-55

    ‘tseries’ is a package for time series analysis and computational
    finance.

    See ‘library(help="tseries")’ for details.
In [ ]:
def buildTable(testName : str, headerList : list[str], justifyList: list[str]) -> Table :
    table = Table(title = testName)
    for i in range(0, len(headerList)) :
        table.add_column(headerList[i], justify = justifyList[i])
    return table

def kpss_test(dataFrameDict : dict[str, pd.DataFrame], colName : str, cryptoList : list, testTarget : str = "") :
    SIGNIFICANCE_LEVEL = 0.05

    kpssTable : Table = buildTable("KPSS" + " " + testTarget, ["Crypto", "StatValue", "pValue", "Outcome"], ["left", "right", "right", "left"])
    for crypto in cryptoList :
        kpssStat, pValue, _, _ = kpss(dataFrameDict[crypto][colName]) ;
        outcomeString = "Reject" if pValue < SIGNIFICANCE_LEVEL else "Accept"
        kpssTable.add_row(crypto, str(kpssStat), str(pValue), outcomeString)
    console = Console()
    console.print(kpssTable)

def adf_test(dataFrameDict : dict[str, pd.DataFrame], colName : str, cryptoList : list, testTarget : str = "") :
    SIGNIFICANCE_LEVEL = 0.05

    adfullerTable = buildTable("ADF" + " " + testTarget, ["Crypto", "StatValue", "pValue", "Outcome"], ["left", "right", "right", "left"])
    for crypto in cryptoList :
        result = adfuller(dataFrameDict[crypto][colName]) ;
        adfullerStat : float = result[0]
        pValue = result[1]
        outcomeString = "Reject" if pValue < SIGNIFICANCE_LEVEL else "Accept"
        adfullerTable.add_row(crypto, str(adfullerStat), str(pValue), outcomeString)
    console = Console()
    console.print(adfullerTable)

def breusch_pagan_test(dataFrameDict : dict[str, pd.DataFrame], colName : str, cryptoList : str, testTarget : str = "", printSummary : bool = False) :
    SIGNIFICANCE_LEVEL = 0.05
    console = Console()
    bpTable = buildTable("Breusch-Pagan", ["Crypto", "StatValue", "pValue", "Outcome"], ["left", "right", "right", "left"])
    for crypto in cryptoList :
        x = sm.add_constant(dataFrameDict[crypto][colName].index.tolist())
        model = sm.OLS(dataFrameDict[crypto][colName].values, x)
        fittedModel = model.fit()
        if printSummary : 
            console.print(crypto + "\n" + str(fittedModel.summary()))
        
        residKurtosis = kurtosis(fittedModel.resid)

        # In the general description of LM test, Greene mentions that this test exaggerates the significance of results in small or moderately large samples. 
        # In this case the F-statistic is preferable.

        # BREUSCH-PAGAN TEST
        _, _, fValue, f_pvalue = het_breuschpagan(fittedModel.resid, fittedModel.model.exog, residKurtosis > 0.05)
        statValue, pValue = fValue, f_pvalue
        
        outcomeString = "Reject" if pValue < SIGNIFICANCE_LEVEL else "Accept"
        bpTable.add_row(crypto, str(statValue), str(pValue), outcomeString)
    console.print(bpTable)

def white_test(dataFrameDict : dict[str, pd.DataFrame], colName : str, cryptoList : str, testTarget : str = "", printSummary : bool = False) :
    SIGNIFICANCE_LEVEL = 0.05
    console = Console()
    whiteTable = buildTable("White" + " " + testTarget, ["Crypto", "StatValue", "pValue", "Outcome"], ["left", "right", "right", "left"])
    for crypto in cryptoList :
    
        x = sm.add_constant(dataFrameDict[crypto][colName].index.tolist())
        model = sm.OLS(dataFrameDict[crypto][colName].values, x)
        fittedModel = model.fit()
        if printSummary :
            console.print(crypto + "\n" + str(fittedModel.summary()))
        
        _, _, fValue, f_pvalue = het_white(fittedModel.resid, fittedModel.model.exog)
        statValue, pValue = fValue, f_pvalue
        outcomeString = "Reject" if pValue < SIGNIFICANCE_LEVEL else "Accept"
        whiteTable.add_row(crypto, str(statValue), str(pValue), outcomeString)
    
    console.print(whiteTable)

def ljung_box_test(dataFrameDict : dict[str, pd.DataFrame], colName : str, cryptoList : list) :
    N_LAGS = 10
    SIGNIFICANCE_LEVEL = 0.05
    ljungTable = buildTable("Ljung-Box", ["Crypto", "StatValue", "pValue", "Outcome"], ["left", "right", "right", "left"])
    for crypto in cryptoList :
        testResult = acorr_ljungbox(dataFrameDict[crypto][colName], lags = N_LAGS)
        statValue, pValue = testResult["lb_stat"][N_LAGS], testResult["lb_pvalue"][N_LAGS]
        outcomeString = "Reject" if pValue < SIGNIFICANCE_LEVEL else "Accept"
        ljungTable.add_row(crypto, str(statValue), str(pValue), outcomeString)

    console = Console()
    console.print(ljungTable)

def correlograms(
        dataFrame : dict[str, pd.DataFrame], 
        colName : str, 
        cryptoList : list, 
        titleAuto : str, 
        titlePartial : str, 
        zeroAuto : bool = True, 
        yLimsAuto : bool = False,
        zeroPar : bool = True,
        yLimsPar : bool = False
    ) :
    N_LAGS = 10

    fig, ax = plt.subplots(figsize=(20, 15), nrows = len(cryptoList), ncols = 2)
    fig.tight_layout(h_pad = 2, w_pad =2)
    i = 0 
    for crypto in cryptoList :
        plot_acf(x = dataFrame[crypto][colName], ax = ax[i][0], lags = N_LAGS, title = crypto + " " + titleAuto, zero = zeroAuto, auto_ylims = yLimsAuto)
        i += 1

    i = 0
    for crypto in cryptoList :
        plot_pacf(x = dataFrame[crypto][colName], ax = ax[i][1], lags = N_LAGS, title = crypto + " " + titlePartial, zero = zeroPar, auto_ylims = yLimsPar)
        i += 1
        
    plt.show()
    plt.clf()

def shapiro_wilk(dataFrameDict : dict[str, pd.DataFrame], colName : str, cryptoList : str, targetName : str = "") :
    # Esegue il test di normalità di Shapiro-Wilk
    SIGNIFICANCE_LEVEL = 0.05
    shapiroTable = buildTable("Shapiro-Wilk " + targetName, ["Crypto", "StatValue", "pValue", "Outcome"], ["left", "right", "right", "left"])
    for crypto in cryptoList:   
        statValue, pValue = shapiro(dataFrameDict[crypto][colName])
        outcomeString = "Reject" if pValue < SIGNIFICANCE_LEVEL else "Accept"
        shapiroTable.add_row(crypto, str(statValue), str(pValue), outcomeString)
    console = Console()
    console.print(shapiroTable)

def kolmogorov_smirnov(dataFrameDict : dict[str, pd.DataFrame], colName : str, cryptoList : str, targetName : str = "") :
    SIGNIFICANCE_LEVEL = 0.05
    ksTable = buildTable("Kolmogorov-Smirnov " + targetName, ["Crypto", "StatValue", "pValue", "Outcome"], ["left", "right", "right", "left"])
    for crypto in cryptoList:   
        statValue, pValue = kstest(dataFrameDict[crypto][colName], norm.cdf)
        outcomeString = "Reject" if pValue < SIGNIFICANCE_LEVEL else "Accept"
        ksTable.add_row(crypto, str(statValue), str(pValue), outcomeString)
    console = Console()
    console.print(ksTable)

Analisi Fianziaria di Criptovalute¶

0. Analisi del Mercato¶

...

1. Estrazione dei Dati¶

I dati sono stati estratti dal sito Yahoo Finance usando delle REST API. I dati sono stati scaricati per tre criptovalute:

  • Ethereum
  • Bitcoin
  • Dash
  • XRP

I dati sono stati scaricati per l'intervallo temporale tra 2018-01-01 e 2023-10-01

In [ ]:
cryptoList : list[str] = ["ETH-USD", "BTC-USD", "DASH-USD", "XRP-USD"]
In [ ]:
extractionStartDate : str = "2018-01-01"
extractionEndDate : str = "2023-09-01"
interval : str = "daily"

for crypto in cryptoList :
    DataExtractor.extractData(crypto, extractionStartDate, extractionEndDate, interval)
In [ ]:
cryptoDataFrameDict : dict[str, pd.DataFrame] = {}
for crypto in cryptoList :
    inputFileName : str = crypto + "_Data.csv"
    inputFilePath = os.path.join("./", "data", inputFileName)
    dataFrame : pd.DataFrame = pd.read_csv(filepath_or_buffer = inputFilePath)
    cryptoDataFrameDict[crypto] = dataFrame

Per tenere conto dell'inflazione degli ultimi anni consideriamo una correzione sui prezzi fatta usando il Consumer Price Index, ottenuto estraendo i dati dal sito della FRED. I dati estratti da quest'ultimo sito sono mensili, mentre i dati scaricati per le criptovalute sono periodici. Per rendere i dati omogenei procediamo facendo un'interpolazione lineare del CPI, ottenendo per ogni giorno il PCI.

In [ ]:
cpiDataFrame = pd.read_csv(
    os.path.join("./", "data", "ConsumerPriceIndex.csv"),
    date_format = "%Y-%m-%d",
    parse_dates = ["DATE"]
    )
cpiDataFrame["DATE"] = pd.to_datetime(cpiDataFrame["DATE"])
cpiDataFrame
Out[ ]:
DATE CPIAUCSL
0 2017-12-01 247.805
1 2018-01-01 248.859
2 2018-02-01 249.529
3 2018-03-01 249.577
4 2018-04-01 250.227
... ... ...
65 2023-05-01 303.294
66 2023-06-01 303.841
67 2023-07-01 304.348
68 2023-08-01 306.269
69 2023-09-01 307.481

70 rows × 2 columns

In [ ]:
cpiInterpolatedDataFrame : pd.DataFrame = pd.DataFrame(columns = ["Date", "CPI"])

for i in range(0, cpiDataFrame.shape[0] - 1) :
    firstValue = cpiDataFrame.iloc[i]["CPIAUCSL"]
    secondValue = cpiDataFrame.iloc[i + 1]["CPIAUCSL"]
    valueDiff = secondValue - firstValue

    firstDate = cpiDataFrame.iloc[i]["DATE"]
    secondDate = cpiDataFrame.iloc[i + 1]["DATE"]
    dateDiff = secondDate - firstDate

    diffDays = dateDiff.days
    interValue = valueDiff / diffDays

    interpolatedValue = firstValue
    currDate = firstDate
    for dateIncr in range(1, diffDays + 1) :
        newRow = [currDate, interpolatedValue]
        cpiInterpolatedDataFrame.loc[len(cpiInterpolatedDataFrame)] = newRow

        interpolatedValue = interpolatedValue + interValue
        currDate = currDate + pd.Timedelta(days = 1)

lastDate = currDate
lastValue = interpolatedValue

cpiInterpolatedDataFrame.loc[len(cpiInterpolatedDataFrame)] = [lastDate, lastValue]
dateTimeStartDate = datetime.datetime.strptime(extractionStartDate, '%Y-%m-%d')
cpiFirstDate = dateTimeStartDate - datetime.timedelta(days = 1)
cpiInterpolatedDataFrame = cpiInterpolatedDataFrame[cpiInterpolatedDataFrame["Date"] >= datetime.datetime(year = cpiFirstDate.year, month = cpiFirstDate.month, day = cpiFirstDate.day)]
cpiInterpolatedDataFrame.reset_index(drop = True, inplace = True)
cpiInterpolatedDataFrame
Out[ ]:
Date CPI
0 2017-12-31 248.825000
1 2018-01-01 248.859000
2 2018-01-02 248.880613
3 2018-01-03 248.902226
4 2018-01-04 248.923839
... ... ...
2066 2023-08-28 307.324613
2067 2023-08-29 307.363710
2068 2023-08-30 307.402806
2069 2023-08-31 307.441903
2070 2023-09-01 307.481000

2071 rows × 2 columns

In [ ]:
cpiInterpolatedDataFrame["Infation"] = cpiInterpolatedDataFrame["CPI"].pct_change() * 100
cpiInterpolatedDataFrame.dropna(inplace = True)
cpiInterpolatedDataFrame.reset_index(inplace = True, drop = True)
cpiInterpolatedDataFrame
Out[ ]:
Date CPI Infation
0 2018-01-01 248.859000 0.013664
1 2018-01-02 248.880613 0.008685
2 2018-01-03 248.902226 0.008684
3 2018-01-04 248.923839 0.008683
4 2018-01-05 248.945452 0.008683
... ... ... ...
2065 2023-08-28 307.324613 0.012723
2066 2023-08-29 307.363710 0.012722
2067 2023-08-30 307.402806 0.012720
2068 2023-08-31 307.441903 0.012718
2069 2023-09-01 307.481000 0.012717

2070 rows × 3 columns

In [ ]:
## TODO Vedere bene come scontare i valori
for crypto in cryptoList :
    dataFrame : pd.DataFrame = cryptoDataFrameDict[crypto]
    dataFrame["NormalizedClose"] = dataFrame["Close"] * (1 - (cpiInterpolatedDataFrame["Infation"]))

cryptoDataFrameDict["ETH-USD"]
Out[ ]:
Date Open High Low Close Adj Close Volume NormalizedClose
0 2018-01-01 755.757019 782.530029 742.004028 772.640991 772.640991 2595760128 762.083453
1 2018-01-02 772.346008 914.830017 772.346008 884.443970 884.443970 5783349760 876.762752
2 2018-01-03 886.000000 974.471008 868.450989 962.719971 962.719971 5093159936 954.359668
3 2018-01-04 961.713013 1045.079956 946.085999 980.921997 980.921997 6502859776 972.404366
4 2018-01-05 975.750000 1075.390015 956.325012 997.719971 997.719971 6683149824 989.057231
... ... ... ... ... ... ... ... ...
2065 2023-08-28 1657.439819 1659.328735 1627.446777 1652.457397 1652.457397 4855588534 1631.432731
2066 2023-08-29 1652.274170 1742.637329 1639.576172 1729.725708 1729.725708 11304916729 1707.720736
2067 2023-08-30 1729.676636 1730.564697 1697.147217 1705.112183 1705.112183 5023904190 1683.423095
2068 2023-08-31 1705.364502 1720.011963 1634.850952 1645.639160 1645.639160 6593153505 1624.709233
2069 2023-09-01 1645.581299 1653.531738 1603.034180 1628.491211 1628.491211 6104510092 1607.782013

2070 rows × 8 columns

2. Analisi del Dataset¶

In [ ]:

2_1. Passaggio al Logaritmo¶

In [ ]:
def logConvertion(dataFrame : pd.DataFrame, columnName : str) :
    dataFrame["LogClosePrice"] = np.log10(dataFrame[columnName])
In [ ]:
for crypto in cryptoList :
    cryptoDf : pd.DataFrame = cryptoDataFrameDict[crypto]
    logConvertion(cryptoDf, "NormalizedClose")

cryptoDataFrameDict["ETH-USD"]["LogClosePrice"]
Out[ ]:
0       2.882003
1       2.942882
2       2.979712
3       2.987847
4       2.995221
          ...   
2065    3.212569
2066    3.232417
2067    3.226193
2068    3.210776
2069    3.206227
Name: LogClosePrice, Length: 2070, dtype: float64
In [ ]:
colors = ["royalblue", "green", "red", "fuchsia"]
figure, axs = plt.subplots(nrows = len(cryptoList), sharex = True, figsize = (20,10))

xtickPoints = cryptoDataFrameDict["ETH-USD"]["Date"]
xtickPoints = xtickPoints[xtickPoints.str.endswith("01") == 1]
i = 0 
for crypto in cryptoList :
    axs[i].plot(cryptoDataFrameDict[crypto]["LogClosePrice"], color = colors[i % len(colors)])
    axs[i].set_title("Andamento " + crypto)
    axs[i].grid(True)
    i += 1

axs[len(cryptoList) - 1].set_xticks(xtickPoints.index)
axs[len(cryptoList) - 1].set_xticklabels(xtickPoints.values, rotation = 90)

plt.show()
plt.clf()
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>
In [ ]:
correlograms(cryptoDataFrameDict, "LogClosePrice", cryptoList, "Autocorrelogramma", "Partial-Autocorrelogramma", zeroPar = False)
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

2_2. Ricerca di una componenti di Random Walk¶

Le componenti di Random Walk vengono ricercate usando due Test di Ipotesi:

  • ADF
  • KPSS

Nel test di KPSS abbiamo:

  • $H_0$ : La serie è stazionaria attorno a una tendenza deterministica (NON ha componente di Random Walk)
  • $H_1$ : La serie non è stazionaria (HA componente di Random Walk)

Nel test di ADF abbiamo:

  • $H_0$ : La serie ha una radice unitaria (HA componente di Random Walk)
  • $H_1$ : La serie non ha una radice unitaria (NON ha componente di Random Walk)

Nel nostro caso vogliamo:

  • KPSS, Rigettare
  • ADF, Accettare
In [ ]:
kpss_test(cryptoDataFrameDict, "LogClosePrice", cryptoList)
adf_test(cryptoDataFrameDict, "LogClosePrice", cryptoList)

## TODO print(result)
## Conclusione: Trend stocastico presente ma abbastanza debole (per l'ultimo)

                        KPSS                        
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃ pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │  4.630432272815153 │   0.01 │ Reject  │
│ BTC-USD  │  4.820559865672976 │   0.01 │ Reject  │
│ DASH-USD │ 2.7531299544913352 │   0.01 │ Reject  │
│ XRP-USD  │  0.918683154822733 │   0.01 │ Reject  │
└──────────┴────────────────────┴────────┴─────────┘

                                ADF                                
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃               pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ -0.9501313424178288 │   0.7710157360162664 │ Accept  │
│ BTC-USD  │  -0.978510604401754 │   0.7610211585021541 │ Accept  │
│ DASH-USD │ -2.8180938650045064 │   0.0557353087206187 │ Accept  │
│ XRP-USD  │  -3.500868599705032 │ 0.007964814582617451 │ Reject  │
└──────────┴─────────────────────┴──────────────────────┴─────────┘

2_3. Differencing della serie storica e analisi del rumore¶

Procediamo lo studio con l'analisi del rumore.

Passiamo al differencing della serie storica, ottenendo $Y_{t+1} - Y_{t} = W_{t}$ e procedendo con un'analisi del termine $W_{t}$.

Il differencing viene fatto sul risultato del passaggio al logaritmo dei valori originari. Siano:

  • Y, valori logaritmici
  • P, valori originari
  • R, rendimento

Fare il differencing dei valori logaritmici ci permette di fare quanto segue:

$Y_{t+1} - Y_{t} = \log(P_{t+1}) - \log(P_t) = \log(\frac{P_{t+1}}{P_{t}}) = \log(R) = W_t$

Ovvero analizzare se il rendimento ha o meno natura di rumore bianco

In [ ]:
differencingDict : dict[str, pd.DataFrame] = {}
for crypto in cryptoDataFrameDict :
    logPriceSeries : pd.Series = cryptoDataFrameDict[crypto]["LogClosePrice"]
    differencing : pd.Series = logPriceSeries.diff()
    differencing.dropna(inplace = True)
    differencing.reset_index(inplace = True, drop = True)
    differencingDict[crypto] = pd.DataFrame({"Diff" : differencing.values, "QuadDiff" : (differencing.pow(2)).values})
In [ ]:
figure, axs = plt.subplots(nrows = len(cryptoList), sharex = True, figsize = (20,10))

xtickPoints = cryptoDataFrameDict["ETH-USD"]["Date"]
xtickPoints = xtickPoints[xtickPoints.str.endswith("01") == 1]
i = 0 
for crypto in cryptoList :
    axs[i].plot(differencingDict[crypto]["Diff"], color = colors[i % len(colors)])
    axs[i].set_title("Andamento Differencing " + crypto)
    axs[i].grid(True)
    i += 1

axs[len(cryptoList) - 1].set_xticks(xtickPoints.index)
axs[len(cryptoList) - 1].set_xticklabels(xtickPoints.values, rotation = 90)

plt.show()
plt.clf()
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

2_3_1. Analisi Stazionarietà¶

Per l'analisi di stazionarietà eseguiamo due test:

  • KDSS
  • Adfuller

Nel test di KPSS abbiamo:

  • H0 : La serie è stazionaria attorno a una tendenza deterministica
  • H1 : La serie non è stazionaria

Nel test di Adfuller abbiamo:

  • H0 : La serie ha una radice unitaria
  • H1 : La serie non ha una radice unitaria

Nel nostro caso vogliamo:

  • KPSS, Accettare
  • Adfuller, Rigettare
In [ ]:
kpss_test(differencingDict, "Diff", cryptoList)
adf_test(differencingDict, "Diff", cryptoList)

                        KPSS                         
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃ pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 0.23201574524870314 │    0.1 │ Accept  │
│ BTC-USD  │ 0.15674142996559834 │    0.1 │ Accept  │
│ DASH-USD │ 0.13769918829263092 │    0.1 │ Accept  │
│ XRP-USD  │ 0.19808342863898767 │    0.1 │ Accept  │
└──────────┴─────────────────────┴────────┴─────────┘

                                 ADF                                 
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃                 pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ -13.748813873133498 │ 1.0600349812346284e-25 │ Reject  │
│ BTC-USD  │ -31.470467916876196 │                    0.0 │ Reject  │
│ DASH-USD │ -21.645263875919735 │                    0.0 │ Reject  │
│ XRP-USD  │  -46.79501030099321 │                    0.0 │ Reject  │
└──────────┴─────────────────────┴────────────────────────┴─────────┘

Dal test di KPSS e dal test ADF, possiamo concludere che il differencing $W_t$ è stazionario

2_3_2. Analisi Eteroschedasticità¶

Per l'analisi di eteroschedasticità eseguiamo i seguenti test:

  • Breusch-Pagan
  • White

Nel test di Breusch-Pagan abbiamo:

  • $H_0$: I residui sono distribuiti con varianza omogenea (La serie è Omoschedastica)
  • $H_1$: I residui NON sono distribuiti con varianza omogenea (La serie NON è omoschedastica)

Nel test di White abbiamo:

  • $H_0$: La varianza dei residui è costante (La serie è Omoschedastica)
  • $H_1$: La varianza dei residui NON è costante (La serie NON è Omoschedastica)

In questo caso vogliamo:

  • Breusch-Pagan, Accettare
  • White, Accettare
In [ ]:
breusch_pagan_test(differencingDict, "Diff", cryptoList, printSummary = True)
white_test(differencingDict, "Diff", cryptoList)

ETH-USD
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.000
Model:                            OLS   Adj. R-squared:                 -0.000
Method:                 Least Squares   F-statistic:                    0.5481
Date:                Wed, 13 Dec 2023   Prob (F-statistic):              0.459
Time:                        15:51:22   Log-Likelihood:                 5049.8
No. Observations:                2069   AIC:                        -1.010e+04
Df Residuals:                    2067   BIC:                        -1.008e+04
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         -0.0004      0.001     -0.472      0.637      -0.002       0.001
x1          5.746e-07   7.76e-07      0.740      0.459   -9.47e-07     2.1e-06
==============================================================================
Omnibus:                      632.947   Durbin-Watson:                   2.106
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            10358.413
Skew:                          -0.995   Prob(JB):                         0.00
Kurtosis:                      13.779   Cond. No.                     2.39e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.39e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

BTC-USD
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.000
Model:                            OLS   Adj. R-squared:                 -0.000
Method:                 Least Squares   F-statistic:                    0.1443
Date:                Wed, 13 Dec 2023   Prob (F-statistic):              0.704
Time:                        15:51:22   Log-Likelihood:                 5578.7
No. Observations:                2069   AIC:                        -1.115e+04
Df Residuals:                    2067   BIC:                        -1.114e+04
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         -0.0001      0.001     -0.143      0.887      -0.002       0.001
x1          2.283e-07   6.01e-07      0.380      0.704    -9.5e-07    1.41e-06
==============================================================================
Omnibus:                      697.962   Durbin-Watson:                   2.089
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            16730.990
Skew:                          -1.029   Prob(JB):                         0.00
Kurtosis:                      16.778   Cond. No.                     2.39e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.39e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

DASH-USD
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.000
Model:                            OLS   Adj. R-squared:                 -0.000
Method:                 Least Squares   F-statistic:                    0.3637
Date:                Wed, 13 Dec 2023   Prob (F-statistic):              0.547
Time:                        15:51:22   Log-Likelihood:                 4747.7
No. Observations:                2069   AIC:                            -9491.
Df Residuals:                    2067   BIC:                            -9480.
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         -0.0013      0.001     -1.252      0.211      -0.003       0.001
x1          5.417e-07   8.98e-07      0.603      0.547   -1.22e-06     2.3e-06
==============================================================================
Omnibus:                      376.450   Durbin-Watson:                   2.095
Prob(Omnibus):                  0.000   Jarque-Bera (JB):             8823.543
Skew:                          -0.083   Prob(JB):                         0.00
Kurtosis:                      13.116   Cond. No.                     2.39e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.39e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

XRP-USD
                            OLS Regression Results                            
==============================================================================
Dep. Variable:                      y   R-squared:                       0.001
Model:                            OLS   Adj. R-squared:                  0.000
Method:                 Least Squares   F-statistic:                     1.386
Date:                Wed, 13 Dec 2023   Prob (F-statistic):              0.239
Time:                        15:51:22   Log-Likelihood:                 4673.8
No. Observations:                2069   AIC:                            -9344.
Df Residuals:                    2067   BIC:                            -9332.
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
const         -0.0015      0.001     -1.315      0.189      -0.004       0.001
x1          1.096e-06   9.31e-07      1.177      0.239    -7.3e-07    2.92e-06
==============================================================================
Omnibus:                      510.216   Durbin-Watson:                   2.059
Prob(Omnibus):                  0.000   Jarque-Bera (JB):            22254.586
Skew:                           0.337   Prob(JB):                         0.00
Kurtosis:                      19.053   Cond. No.                     2.39e+03
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.39e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

                           Breusch-Pagan                           
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃                pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │  8.567367566736669 │ 0.0034597918568196914 │ Reject  │
│ BTC-USD  │  7.763759935805086 │  0.005379165131515243 │ Reject  │
│ DASH-USD │  2.276765009171112 │    0.1314783730532143 │ Accept  │
│ XRP-USD  │ 1.2467607341644058 │    0.2643018406639988 │ Accept  │
└──────────┴────────────────────┴───────────────────────┴─────────┘

                              White                               
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃               pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │   5.26508181913567 │ 0.005238555530471663 │ Reject  │
│ BTC-USD  │  4.141400487811925 │ 0.016032762971599328 │ Reject  │
│ DASH-USD │  3.666072606577798 │ 0.025743255972081297 │ Reject  │
│ XRP-USD  │ 0.7598677076192619 │   0.4678589731665155 │ Accept  │
└──────────┴────────────────────┴──────────────────────┴─────────┘

2_3_3. Analisi Autocorrelazione¶

Per l'Autocorrelazione consideriamo i seguenti test:

  • Ljung-Box

Per il test Ljung-Box:

  • $H_0$: Non c'è autocorrelazione nei dati. In altre parole, i residui sono indipendenti l'uno dall'altro e non mostrano alcuna struttura di autocorrelazione significativa.
  • $H_1$: I residui non sono indipendenti e mostrano una struttura di autocorrelazione significativa.

Vogliamo quindi :

  • Ljung-Box, Accettare
In [ ]:
ljung_box_test(differencingDict, "Diff", cryptoList)

                             Ljung-Box                             
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃                pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 31.609722534078987 │ 0.0004650065716092994 │ Reject  │
│ BTC-USD  │ 16.925034452734113 │   0.07603932218108504 │ Accept  │
│ DASH-USD │ 15.716773963771859 │   0.10803220218423684 │ Accept  │
│ XRP-USD  │  7.977674132723926 │     0.631017790629593 │ Accept  │
└──────────┴────────────────────┴───────────────────────┴─────────┘

Tracciamo gli autocorrelogrammi e gli autocorrelogrammi parziali dei dati.

In [ ]:
correlograms(differencingDict, "Diff", cryptoList, titleAuto = "Differencing Autocorrelation", titlePartial = "Differencing Partial-Autocorrelation", zeroAuto = False, yLimsAuto = True, zeroPar = False, yLimsPar = True)
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

Il test quindi Accetta per entrambi i casi e i dati non dovrebbero essere autocorrelati. Tuttavia, osservando i grafici relativi al differencing, possiamo notare come vi siano dei cluster di varianza costante, in cui a basse varianze seguono zone di bassa varianza (analogamente per quelle ad alta varianza).

Riassumendo abbiamo:

  • Accettazione del test di Ljung-Box
  • Presenza di clusterizzazione delle varianze

Questo fenomeno potrebbe essere dovuto al fatto che i dati sono molti, e quindi il grande numero di dati potrebbe offuscare l'autocorrelazione portando i dati a tendere al gaussiano.

Ci fa supporre la presenza di punti di rottura nella struttura.

Passiamo quindi, dopo dei veloci test sulla gaussianità, all'analisi delle rotture strutturali.

2_3_4. Analisi Normalità¶

Per l'analisi di normalità eseguiamo i seguenti test:

  • Shapiro-Wilk
  • Kolmogorov-Smirnov

Nel test di Shapiro-Wilk abbiamo:

  • $H_0$: I dati seguono una distribuzione normale
  • $H_1$: I dati NON seguono una distribuzione normale

Nel test di Kolmogorov-Smirnov abbiamo:

  • $H_0$: I dati seguono la distribuzione specificata
  • $H_1$: I dati NON seguono la distribuzione specificata

Nel nostro caso vogliamo:

  • Shapiro-Wilk, Accettare
  • Kolmogorov-Smirnov, Accettare
In [ ]:
shapiro_wilk(differencingDict, "Diff", cryptoList)
kolmogorov_smirnov(differencingDict, "Diff", cryptoList)

                            Shapiro-Wilk                            
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃                 pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 0.9225029349327087 │  2.243272511905845e-31 │ Reject  │
│ BTC-USD  │ 0.9065624475479126 │  7.687562699532387e-34 │ Reject  │
│ DASH-USD │ 0.9040272235870361 │ 3.3494620980164206e-34 │ Reject  │
│ XRP-USD  │ 0.8357081413269043 │   7.15923385423549e-42 │ Reject  │
└──────────┴────────────────────┴────────────────────────┴─────────┘

                 Kolmogorov-Smirnov                  
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃ pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │  0.4705378122861261 │    0.0 │ Reject  │
│ BTC-USD  │ 0.47608764853183083 │    0.0 │ Reject  │
│ DASH-USD │ 0.46608875744466827 │    0.0 │ Reject  │
│ XRP-USD  │  0.4642030732172747 │    0.0 │ Reject  │
└──────────┴─────────────────────┴────────┴─────────┘

Notiamo quindi che i due test ci fanno rigettare in entrambi i casi l'ipotesi nulla di normalità dei dati. Per completezza tracciamo un grafico dei dati.

2_4. Analisi dei Punti di Rottura nella Struttura¶

Cerchiamo i punti di rottura usando la funzione breakpoints del package strucchange di R.

In [ ]:
# Loading the function we have defined in R.
r['source']('./R/R_Scripts.R')

breakPointSearch_R = robjects.globalenv['breakPointSearch']
fstatBreakpointsSearch_R = robjects.globalenv['fstatBreakpointsSearch']
supmzBreakpointsSearch_R = robjects.globalenv['supmzBreakpointsSearch']

cryptoBreakPointsDict_1 = {}
cryptoBreakPointsDict_2 = {}
cryptoBreakPointsDict_3 = {}

for crypto in cryptoList :
    ## Converting to R dataFrame
    closePrice_R = pandas2ri.py2rpy(cryptoDataFrameDict[crypto]["LogClosePrice"])

    # Invoking R function
    breakPointsList_R = breakPointSearch_R(closePrice_R)

    #Converting it back to a pandas dataframe.
    breakPointsList = np.array(breakPointsList_R)

    breakPointsList = [int(elem - 1) for elem in breakPointsList]
    cryptoBreakPointsDict_1[crypto] = breakPointsList

    fstatPointsList_R = fstatBreakpointsSearch_R(closePrice_R)
    fstatBreakPointsList = np.array(fstatPointsList_R)
    fstatBreakPointsList = [int(elem - 1) for elem in fstatBreakPointsList]
    cryptoBreakPointsDict_2[crypto] = fstatBreakPointsList

    logClosePriceDataFrame = pd.DataFrame(cryptoDataFrameDict[crypto]["LogClosePrice"], columns = ["LogClosePrice"])
    logClosePriceDF_R = pandas2ri.py2rpy(logClosePriceDataFrame)
    supmzPointsList_R = supmzBreakpointsSearch_R(logClosePriceDF_R)
    supmzBreakPointsList = list(supmzPointsList_R)
    cryptoBreakPointsDict_3[crypto] = supmzBreakPointsList
In [ ]:
breakPointsTable = buildTable("Structural Breakpoints", ["Crypto", "Breakpoint", "FStats", "SupMZ"], ["left", "right", "right", "right"])
for crypto in cryptoList :
    datesString_1 = "| "
    datesString_2 = "| "
    datesString_3 = "| "
    for breakpoint in cryptoBreakPointsDict_1[crypto] :
        date = cryptoDataFrameDict[crypto]["Date"].loc[breakpoint]
        datesString_1 += date + " | "
    for breakpoint in cryptoBreakPointsDict_2[crypto] :
        date = cryptoDataFrameDict[crypto]["Date"].loc[breakpoint]
        datesString_2 += date + " | "
    for breakpoint in cryptoBreakPointsDict_3[crypto] :
        date = cryptoDataFrameDict[crypto]["Date"].loc[breakpoint]
        datesString_3 += date + " | "
    breakPointsTable.add_row(crypto, datesString_1, datesString_2, datesString_3)

console = Console()
console.print(breakPointsTable)

                                         Structural Breakpoints                                          
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━┓
┃ Crypto   ┃                                             Breakpoint ┃          FStats ┃           SupMZ ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━┩
│ ETH-USD  │  | 2018-11-06 | 2020-02-06 | 2021-01-02 | 2022-05-10 | │  | 2020-12-29 | │  | 2021-01-04 | │
│ BTC-USD  │               | 2019-05-12 | 2020-12-15 | 2022-05-08 | │  | 2020-11-11 | │  | 2020-11-06 | │
│ DASH-USD │  | 2018-11-06 | 2019-09-22 | 2021-01-09 | 2022-05-04 | │  | 2018-11-07 | │  | 2022-05-12 | │
│ XRP-USD  │  | 2018-11-18 | 2019-09-24 | 2021-02-08 | 2022-04-28 | │  | 2021-02-08 | │  | 2021-02-05 | │
└──────────┴────────────────────────────────────────────────────────┴─────────────────┴─────────────────┘
In [ ]:
colors = ["blue", "green", "red", "fuchsia"]

figure, axs = plt.subplots(nrows = len(cryptoList), sharex = True, figsize = (20,10))

xtickPoints = cryptoDataFrameDict["ETH-USD"]["Date"]
xtickPoints = xtickPoints[xtickPoints.str.endswith("01") == 1]
i = 0 
legend_1 = True
legend_2 = True
legend_3 = True
for crypto in cryptoList :
    axs[i].plot(cryptoDataFrameDict[crypto]["LogClosePrice"], color = colors[i % len(colors)])
    axs[i].set_title("Breakpoints per " + crypto)
    # axs[i].grid(True)

    for breakpoint in cryptoBreakPointsDict_1[crypto] :
        axs[i].axvline(x = breakpoint, color = "black", linestyle = "--", linewidth = "2", label = "Breakpoints" if legend_1 else None)
        legend_1 = False
    for breakpoint in cryptoBreakPointsDict_2[crypto] :
        axs[i].axvline(x = breakpoint, color = "orange", linestyle = "--", linewidth = "2", label = "FStats" if legend_2 else None)
        legend_2 = False
    for breakpoint in cryptoBreakPointsDict_3[crypto] :
        axs[i].axvline(x = breakpoint, color = "gray", linestyle = "--", linewidth = "2", label = "SupMZ" if legend_3 else None)
        legend_3 = False
    i += 1
figure.legend()

axs[len(cryptoList) - 1].set_xticks(xtickPoints.index)
axs[len(cryptoList) - 1].set_xticklabels(xtickPoints.values, rotation = 90)

plt.show()
plt.clf()


figure, axs = plt.subplots(nrows = len(cryptoList), sharex = True, figsize = (20,10))

xtickPoints = cryptoDataFrameDict["ETH-USD"]["Date"]
xtickPoints = xtickPoints[xtickPoints.str.endswith("01") == 1]
i = 0
legend_1 = True
legend_2 = True
legend_3 = True
for crypto in cryptoList :
    axs[i].plot(differencingDict[crypto]["Diff"], color = colors[i % len(colors)])
    axs[i].set_title("Andamento Differencing " + crypto)
    axs[i].grid(True)

    for breakpoint in cryptoBreakPointsDict_1[crypto] :
        axs[i].axvline(x = breakpoint, color = "black", linestyle = "--", linewidth = "2", label = "Breakpoints" if legend_1 else None)
        legend_1 = False
    for breakpoint in cryptoBreakPointsDict_2[crypto] :
        axs[i].axvline(x = breakpoint, color = "orange", linestyle = "--", linewidth = "2", label = "FStats" if legend_2 else None)
        legend_2 = False
    for breakpoint in cryptoBreakPointsDict_3[crypto] :
        axs[i].axvline(x = breakpoint, color = "gray", linestyle = "--", linewidth = "2", label = "SupMZ" if legend_3 else None)
        legend_3 = False
        
    i += 1
figure.legend()

axs[len(cryptoList) - 1].set_xticks(xtickPoints.index)
axs[len(cryptoList) - 1].set_xticklabels(xtickPoints.values, rotation = 90)

plt.show()
plt.clf()
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

I test rilevano quindi la presenza di punti di rottura.

Notiamo in particolare che test diversi rilevano punti di rottura simili: scegliamo quindi di procedere con un "voto a maggioranza", prendendo il punto di rottura più simile tra i tre test.

3. Divisione del dataset¶

Dividiamo il dataset in due parti, training e test set: manteniamo nel training circa il 90% dei dati di ogni strumento

In [ ]:
cryptoBreakPointsIndexes = {
    "ETH-USD" : cryptoBreakPointsDict_1["ETH-USD"][2],
    "BTC-USD" : cryptoBreakPointsDict_1["BTC-USD"][1],
    "DASH-USD" : cryptoBreakPointsDict_1["DASH-USD"][3],
    "XRP-USD" : cryptoBreakPointsDict_1["XRP-USD"][2],
}
In [ ]:
restrCryptoDataFrameDict : dict[str, pd.DataFrame] = {}
restrDiffDataFrameDict : dict[str, pd.DataFrame] = {}

for crypto in cryptoList :
    breakPoint = cryptoBreakPointsIndexes[crypto]

    restrictedDataFrame : pd.DataFrame = cryptoDataFrameDict[crypto].iloc[cryptoDataFrameDict[crypto].index >= breakPoint]
    restrictedDataFrame.reset_index(drop = True, inplace = True)
    restrCryptoDataFrameDict[crypto] = restrictedDataFrame

    restrictedDiffDataFrame : pd.DataFrame = differencingDict[crypto].iloc[differencingDict[crypto]["Diff"].index >= breakPoint]
    restrictedDiffDataFrame.reset_index(drop = True, inplace = True)
    restrDiffDataFrameDict[crypto] = restrictedDiffDataFrame
In [ ]:
trainCryptoDict : dict[str, pd.DataFrame] = {}
testCryptoDict : dict[str, pd.DataFrame] = {}
trainCryptoDiffDict : dict[str, pd.DataFrame] = {}
testCryptoDiffDict : dict[str, pd.DataFrame] = {}

for crypto in cryptoList :
    totalSize = restrCryptoDataFrameDict[crypto].shape[0]
    trainSize = int((totalSize / 100) * 90)
    testSize = totalSize - trainSize

    trainCryptoDict[crypto] = restrCryptoDataFrameDict[crypto].iloc[restrCryptoDataFrameDict[crypto].index < trainSize]
    trainCryptoDict[crypto].reset_index(drop = True, inplace = True)
    testCryptoDict[crypto] = restrCryptoDataFrameDict[crypto].iloc[restrCryptoDataFrameDict[crypto].index >= trainSize]
    testCryptoDict[crypto].reset_index(drop = True, inplace = True)

    trainCryptoDiffDict[crypto] = restrDiffDataFrameDict[crypto].iloc[restrDiffDataFrameDict[crypto].index < trainSize]
    trainCryptoDiffDict[crypto].reset_index(drop = True, inplace = True)
    testCryptoDiffDict[crypto] = restrDiffDataFrameDict[crypto].iloc[restrDiffDataFrameDict[crypto].index >= trainSize]
    testCryptoDiffDict[crypto].reset_index(drop = True, inplace = True)

4. Analisi dei dati ristretti¶

In [ ]:
colors = ["blue", "green", "red", "fuchsia"]
figure, axs = plt.subplots(nrows = len(cryptoList), ncols = 3, figsize = (20,15))
figure.tight_layout(h_pad = 7.5, w_pad = 2)

STEP = 45
ROTATION = 90

i = 0 
for crypto in cryptoList :
    axs[i][0].plot(trainCryptoDict[crypto]["LogClosePrice"], color = colors[i % len(colors)])
    axs[i][0].set_title("Andamento Restrizione " + crypto)
    axs[i][0].grid(True)

    xtickPoints = np.arange(start = 0, stop = trainCryptoDict[crypto].shape[0], step = STEP)
    xtickLabels = trainCryptoDict[crypto]["Date"].iloc[xtickPoints]
    axs[i][0].set_xticks(xtickPoints)
    axs[i][0].set_xticklabels(xtickLabels, rotation = ROTATION)

    i += 1

i = 0 
for crypto in cryptoList :
    axs[i][1].plot(trainCryptoDiffDict[crypto]["Diff"], color = colors[i % len(colors)])
    axs[i][1].set_title("Differencing Restrizione " + crypto)
    axs[i][1].grid(True)

    xtickPoints = np.arange(start = 0, stop = trainCryptoDict[crypto].shape[0], step = STEP)
    xtickLabels = trainCryptoDict[crypto]["Date"].iloc[xtickPoints]
    axs[i][1].set_xticks(xtickPoints)
    axs[i][1].set_xticklabels(xtickLabels, rotation = ROTATION)

    i += 1

i = 0 
for crypto in cryptoList :
    axs[i][2].plot(trainCryptoDiffDict[crypto]["QuadDiff"], color = colors[i % len(colors)])
    axs[i][2].set_title("Quadratic Differencing Restrizione " + crypto)
    axs[i][2].grid(True)

    xtickPoints = np.arange(start = 0, stop = trainCryptoDict[crypto].shape[0], step = STEP)
    xtickLabels = trainCryptoDict[crypto]["Date"].iloc[xtickPoints]
    axs[i][2].set_xticks(xtickPoints)
    axs[i][2].set_xticklabels(xtickLabels, rotation = ROTATION)

    i += 1

plt.show()
plt.clf()
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

4_1. Analisi del differencing ridotto¶

Come nel primo caso analizziamo il differencing e vediamo se questo ha natura di white noise.

4_1_1. Analisi Stazionarietà¶

Per l'analisi di stazionarietà eseguiamo due test:

  • KDSS
  • Adfuller

Nel test di KPSS abbiamo:

  • H0 : La serie è stazionaria attorno a una tendenza deterministica
  • H1 : La serie non è stazionaria

Nel test di Adfuller abbiamo:

  • H0 : La serie ha una radice unitaria
  • H1 : La serie non ha una radice unitaria

Nel nostro caso vogliamo:

  • KPSS, Accettare
  • Adfuller, Rigettare
In [ ]:
kpss_test(trainCryptoDiffDict, "Diff", cryptoList)
adf_test(trainCryptoDiffDict, "Diff", cryptoList)

                        KPSS                         
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃ pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 0.30402339982507376 │    0.1 │ Accept  │
│ BTC-USD  │ 0.26793706713609056 │    0.1 │ Accept  │
│ DASH-USD │ 0.14448487432870857 │    0.1 │ Accept  │
│ XRP-USD  │ 0.10857489114721602 │    0.1 │ Accept  │
└──────────┴─────────────────────┴────────┴─────────┘

                                ADF                                 
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃                pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ -31.463043189797844 │                   0.0 │ Reject  │
│ BTC-USD  │ -30.483765580661746 │                   0.0 │ Reject  │
│ DASH-USD │ -15.051422863190458 │ 9.281783031352217e-28 │ Reject  │
│ XRP-USD  │ -31.024464015415518 │                   0.0 │ Reject  │
└──────────┴─────────────────────┴───────────────────────┴─────────┘
4_1_2. Analisi Eteroschedasticità¶

Per l'analisi di eteroschedasticità eseguiamo i seguenti test:

  • Breusch-Pagan
  • White

Nel test di Breusch-Pagan abbiamo:

  • $H_0$: I residui sono distribuiti con varianza omogenea (La serie è Omoschedastica)
  • $H_1$: I residui NON sono distribuiti con varianza omogenea (La serie NON è omoschedastica)

Nel test di White abbiamo:

  • $H_0$: La varianza dei residui è costante (La serie è Omoschedastica)
  • $H_1$: La varianza dei residui NON è costante (La serie NON è Omoschedastica)

In questo caso vogliamo:

  • Breusch-Pagan, Accettare
  • White, Accettare
In [ ]:
breusch_pagan_test(trainCryptoDiffDict, "Diff", cryptoList)
white_test(trainCryptoDiffDict, "Diff", cryptoList)

                           Breusch-Pagan                           
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃         StatValue ┃                 pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 25.05566267671057 │  6.737702332420841e-07 │ Reject  │
│ BTC-USD  │ 25.84501763683427 │ 4.5114279068198483e-07 │ Reject  │
│ DASH-USD │ 2.495168564366106 │    0.11492299248948996 │ Accept  │
│ XRP-USD  │  34.8344311264828 │  5.204886672228533e-09 │ Reject  │
└──────────┴───────────────────┴────────────────────────┴─────────┘

                              White                               
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃         StatValue ┃                pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │  13.1651660977433 │  2.32843202022156e-06 │ Reject  │
│ BTC-USD  │ 13.22301453319326 │ 2.193663541133534e-06 │ Reject  │
│ DASH-USD │ 3.299977334208391 │  0.037811687468737934 │ Reject  │
│ XRP-USD  │ 23.00017453495107 │ 1.885248700990853e-10 │ Reject  │
└──────────┴───────────────────┴───────────────────────┴─────────┘
4_1_3. Analisi Autocorrelazione¶

Per l'Autocorrelazione consideriamo i seguenti test:

  • Ljung-Box

Per il test Ljung-Box:

  • $H_0$: Non c'è autocorrelazione nei dati. In altre parole, i residui sono indipendenti l'uno dall'altro e non mostrano alcuna struttura di autocorrelazione significativa.
  • $H_1$: I residui non sono indipendenti e mostrano una struttura di autocorrelazione significativa.

Vogliamo quindi :

  • Ljung-Box, Accettare
In [ ]:
ljung_box_test(trainCryptoDiffDict, "Diff", cryptoList)

                            Ljung-Box                             
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃               pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │  20.74439638246904 │ 0.022948121087501706 │ Reject  │
│ BTC-USD  │ 12.632829754955273 │  0.24493117060834974 │ Accept  │
│ DASH-USD │  9.348220225953458 │  0.49940562393741117 │ Accept  │
│ XRP-USD  │ 16.883011215521183 │  0.07699305871115295 │ Accept  │
└──────────┴────────────────────┴──────────────────────┴─────────┘
In [ ]:
correlograms(
    trainCryptoDiffDict,
    "Diff",
    cryptoList,
    titleAuto = "Restricted Differencing Autocorrelation",
    titlePartial = "Restricted Differencing Partial-Autocorrelation", 
    zeroAuto = False, yLimsAuto = True,
    zeroPar = False, yLimsPar = True
)
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

Osserviamo forte correlazione (effetto clustering) e quindi potremmo procedere a modellarli con dei processi GARCH.

3_1_4. Analisi Normalità¶

Per l'analisi di normalità eseguiamo i seguenti test:

  • Shapiro-Wilk
  • Kolmogorov-Smirnov

Nel test di Shapiro-Wilk abbiamo:

  • $H_0$: I dati seguono una distribuzione normale
  • $H_1$: I dati NON seguono una distribuzione normale

Nel test di Kolmogorov-Smirnov abbiamo:

  • $H_0$: I dati seguono la distribuzione specificata
  • $H_1$: I dati NON seguono la distribuzione specificata

Nel nostro caso vogliamo:

  • Shapiro-Wilk, Accettare
  • Kolmogorov-Smirnov, Accettare
In [ ]:
shapiro_wilk(trainCryptoDiffDict, "Diff", cryptoList)
kolmogorov_smirnov(trainCryptoDiffDict, "Diff", cryptoList)

                            Shapiro-Wilk                            
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃                 pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 0.9502420425415039 │ 1.3270286513620812e-16 │ Reject  │
│ BTC-USD  │ 0.9572618007659912 │  1.919576134737323e-15 │ Reject  │
│ DASH-USD │  0.939193844795227 │ 2.2313227654446877e-12 │ Reject  │
│ XRP-USD  │ 0.8953757286071777 │ 1.7658039207309898e-23 │ Reject  │
└──────────┴────────────────────┴────────────────────────┴─────────┘

                          Kolmogorov-Smirnov                          
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃                  pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 0.47223125234355756 │ 1.3485745072511574e-179 │ Reject  │
│ BTC-USD  │  0.4785933494051804 │ 3.9304122976977064e-188 │ Reject  │
│ DASH-USD │  0.4741664051569694 │   9.750037391596386e-91 │ Reject  │
│ XRP-USD  │ 0.46522174716623627 │ 1.8352320088248526e-167 │ Reject  │
└──────────┴─────────────────────┴─────────────────────────┴─────────┘

5. Quadratic autocorrelation¶

Analizziamo l'autocorrelazione dei quadrati: osservando i grafici possiamo infatti notare come a periodi di alta volatilità seguano periodi di alta volatilità e come a periodi di bassa volatilità seguano periodi di bassa volatilità. Questo ci fa supporre la presenza di una autocorrelazione nella serie delle varianze.

Ricordando che:

$D^{2}[X_t] = E[X_t^{2}] + E^{2}[X_t]$

Se la media fosse nulla quindi la varianza dei rendimenti corrisponderebbe con il quadrato dei rendimenti stessi:

$D^{2}[X_t] = E[X_t^{2}]$

In [ ]:
#TODO media nulla -> interessante (indice di mercato efficiente) (BTC prof)
# Ipotesi nulla: la media è zero
null_hypothesis_mean = 0
alpha = 0.05
meanTable = buildTable("Zero Mean Test", ["Crypto", "T-Statistic", "P-Value", "Result"], ["left", "right", "right", "left"])
for crypto in cryptoList:
    # Esegue il test t
    t_statistic, p_value = ttest_1samp(differencingDict[crypto]["Diff"], null_hypothesis_mean)

    # Confronta il p-value con un livello di significatività predefinito (ad esempio, 0.05)
    outcomeString = "Reject" if p_value < alpha else "Accept"
    meanTable.add_row(crypto, str(t_statistic), str(p_value), outcomeString)
# Stampa i risultati del test
console = Console()
console.print(meanTable)

                         Zero Mean Test                          
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┓
┃ Crypto   ┃         T-Statistic ┃             P-Value ┃ Result ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━┩
│ ETH-USD  │ 0.33808066325759595 │  0.7353367184851297 │ Accept │
│ BTC-USD  │  0.3725821323911949 │  0.7094976528285228 │ Accept │
│ DASH-USD │  -1.459006273766798 │ 0.14471524793122822 │ Accept │
│ XRP-USD  │ -0.5918757465602607 │  0.5539984936921993 │ Accept │
└──────────┴─────────────────────┴─────────────────────┴────────┘

Per nessuno degli strumenti finanziari possiamo rigettare l'ipotesi nulla di media nulla al 95%.

Procediamo quindi con l'analisi dei rendimenti logaritmici quadrati. Analizziamo l'eventuale presenza di autocorrelazione con il test di Ljung-Box.

In [ ]:
ljung_box_test(trainCryptoDiffDict, "QuadDiff", cryptoList)

                             Ljung-Box                              
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃                 pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 107.45662340504884 │ 1.7362744278257656e-18 │ Reject  │
│ BTC-USD  │ 28.435878927917237 │  0.0015366636984585028 │ Reject  │
│ DASH-USD │ 20.296369471575147 │   0.026570915198950266 │ Reject  │
│ XRP-USD  │ 124.35893071112875 │  6.585066550321394e-22 │ Reject  │
└──────────┴────────────────────┴────────────────────────┴─────────┘

Dai risultati si evince che per tutte le monete si può rigettare l'ipotesi nulla di non-autocorrelazione, concludiamo quindi che tutte le monete presentano autocorrelazione.

Procediamo a graficare quindi gli autocorrelogrammi per le varie monete.

In [ ]:
correlograms(
    trainCryptoDiffDict, 
    "QuadDiff", 
    cryptoList,
    titleAuto = "Restricted Differencing (Quadratic) Autocorrelation",
    titlePartial = "Restricted Differencing (Quadratic) Partial-Autocorrelation",
    zeroAuto = False, zeroPar = False,
    yLimsAuto = True, yLimsPar = True
)
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

6. Analisi con modello GARCH¶

Essendo presente autocorrelazione nella volatilità dei rendimenti possiamo procedere a modellarla tramite un Generalized ARCH model (GARCH).

Nel modello GARCH ...In practice, only rather rich ARCH parameterizations are able to fit financial series adequately However, largely parameterized models can be unstable in forecasting and a hard to estimate In order to overcome the shortcomings of the ARCH, Tim Bollerslev proposed a generalisation of the ARCH model called GARCH (Bollerslev, 1986) The model allows to fit financial returns adequately while keeping the number of parameters small In practice, the GARCH model is one of the most successfully employed volatility models

(da pag 63 a 110 circa)

In [ ]:
residualsDict : {str, pd.DataFrame} = {}

r['source']('./R/R_Scripts.R')
garchResiduals_R = robjects.globalenv['garchResiduals']

for crypto in cryptoList :
    tempCryptoDiff = trainCryptoDiffDict[crypto]["Diff"].copy()
    #tempCryptoDiff = differencingDict[crypto]["Diff"].copy()
    model = arch_model(tempCryptoDiff)
    fittedModel = model.fit(disp='off')  # Setting disp to 'off' to suppress convergence messages
    console.print(crypto + "\n" + str(fittedModel.summary()))

    diffList_R = pandas2ri.py2rpy(tempCryptoDiff)
    residuals_R = garchResiduals_R(diffList_R)
    residuals = pd.Series(list(residuals_R)[1 : ]).dropna()

    diff = residuals - fittedModel.resid

    residualsDict[crypto] = pd.DataFrame(data = {"Resid" : residuals})

ETH-USD
                     Constant Mean - GARCH Model Results                      
==============================================================================
Dep. Variable:                   Diff   R-squared:                       0.000
Mean Model:             Constant Mean   Adj. R-squared:                  0.000
Vol Model:                      GARCH   Log-Likelihood:                2214.67
Distribution:                  Normal   AIC:                          -4421.35
Method:            Maximum Likelihood   BIC:                          -4402.25
                                        No. Observations:                  875
Date:                Wed, Dec 13 2023   Df Residuals:                      874
Time:                        15:54:33   Df Model:                            1
                                  Mean Model                                 
=============================================================================
                 coef    std err          t      P>|t|       95.0% Conf. Int.
-----------------------------------------------------------------------------
mu         5.1548e-04  5.980e-04      0.862      0.389 [-6.566e-04,1.688e-03]
                              Volatility Model                              
============================================================================
                 coef    std err          t      P>|t|      95.0% Conf. Int.
----------------------------------------------------------------------------
omega      8.7622e-06  3.227e-11  2.715e+05      0.000 [8.762e-06,8.762e-06]
alpha[1]       0.0999  3.855e-02      2.592  9.552e-03   [2.435e-02,  0.175]
beta[1]        0.8801  3.175e-02     27.722 3.824e-169     [  0.818,  0.942]
============================================================================

Covariance estimator: robust

 ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 


     I     INITIAL X(I)        D(I)

     1     3.927254e-04     1.000e+00
     2     5.000000e-02     1.000e+00
     3     5.000000e-02     1.000e+00

    IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
     0    1 -2.964e+03
     1    7 -2.964e+03  1.76e-04  2.76e-04  1.6e-04  3.4e+09  1.6e-05  4.66e+05
     2    8 -2.965e+03  1.17e-05  1.29e-05  1.4e-04  2.0e+00  1.6e-05  1.27e+00
     3   15 -2.967e+03  8.08e-04  1.17e-03  2.8e-01  2.0e+00  4.0e-02  1.26e+00
     4   18 -2.975e+03  2.58e-03  7.88e-03  8.4e-01  1.9e+00  6.4e-01  2.74e-01
     5   30 -2.999e+03  8.07e-03  5.29e-02  3.9e-05  4.0e+00  5.4e-05  9.59e-02
     6   36 -3.005e+03  1.98e-03  1.48e-03  9.8e-03  1.8e+00  1.4e-02  1.33e-02
     7   37 -3.005e+03  1.83e-04  2.71e-03  9.9e-03  1.9e+00  1.4e-02  1.23e-02
     8   38 -3.010e+03  1.40e-03  2.52e-03  4.7e-03  1.9e+00  6.8e-03  2.72e-03
     9   41 -3.012e+03  6.81e-04  8.80e-04  5.4e-02  5.2e-01  7.9e-02  1.00e-03
    10   42 -3.014e+03  7.13e-04  1.36e-03  3.5e-02  4.3e-01  7.9e-02  1.44e-03
    11   44 -3.017e+03  1.22e-03  2.02e-03  1.8e-02  1.3e+00  3.1e-02  2.84e-03
    12   45 -3.018e+03  2.68e-04  4.34e-04  2.2e-02  0.0e+00  4.1e-02  4.34e-04
    13   46 -3.019e+03  1.95e-04  3.54e-04  1.9e-02  8.7e-01  4.1e-02  4.53e-04
    14   47 -3.019e+03  8.44e-05  1.20e-04  9.9e-03  0.0e+00  2.1e-02  1.20e-04
    15   48 -3.019e+03  2.03e-05  1.91e-05  5.9e-03  0.0e+00  1.3e-02  1.91e-05
    16   49 -3.019e+03  5.28e-07  6.36e-07  4.6e-04  0.0e+00  8.4e-04  6.36e-07
    17   50 -3.019e+03  2.35e-08  3.30e-08  7.4e-06  0.0e+00  1.8e-05  3.30e-08
    18   51 -3.019e+03  3.44e-09  3.71e-09  9.9e-06  1.5e+00  1.8e-05  4.38e-09
    19   52 -3.019e+03  6.61e-10  5.34e-10  1.4e-05  0.0e+00  2.6e-05  5.34e-10
    20   53 -3.019e+03 -2.56e-11  7.08e-12  1.6e-06  0.0e+00  2.9e-06  7.08e-12

 ***** RELATIVE FUNCTION CONVERGENCE *****

 FUNCTION    -3.019151e+03   RELDX        1.606e-06
 FUNC. EVALS      53         GRAD. EVALS      20
 PRELDF       7.076e-12      NPRELDF      7.076e-12

     I      FINAL X(I)        D(I)          G(I)

     1    1.092235e-05     1.000e+00    -4.028e+01
     2    9.746033e-02     1.000e+00     7.699e-03
     3    8.785455e-01     1.000e+00    -1.044e-03


BTC-USD
                     Constant Mean - GARCH Model Results                      
==============================================================================
Dep. Variable:                   Diff   R-squared:                       0.000
Mean Model:             Constant Mean   Adj. R-squared:                  0.000
Vol Model:                      GARCH   Log-Likelihood:                2457.09
Distribution:                  Normal   AIC:                          -4906.17
Method:            Maximum Likelihood   BIC:                          -4887.00
                                        No. Observations:                  891
Date:                Wed, Dec 13 2023   Df Residuals:                      890
Time:                        15:54:33   Df Model:                            1
                                  Mean Model                                 
=============================================================================
                 coef    std err          t      P>|t|       95.0% Conf. Int.
-----------------------------------------------------------------------------
mu         1.5056e-04  4.904e-04      0.307      0.759 [-8.106e-04,1.112e-03]
                              Volatility Model                              
============================================================================
                 coef    std err          t      P>|t|      95.0% Conf. Int.
----------------------------------------------------------------------------
omega      5.2742e-06  1.534e-13  3.437e+07      0.000 [5.274e-06,5.274e-06]
alpha[1]       0.0500  8.641e-04     57.854      0.000 [4.830e-02,5.169e-02]
beta[1]        0.9298  5.429e-03    171.285      0.000     [  0.919,  0.940]
============================================================================

Covariance estimator: robust

 ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 


     I     INITIAL X(I)        D(I)

     1     2.265060e-04     1.000e+00
     2     5.000000e-02     1.000e+00
     3     5.000000e-02     1.000e+00

    IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
     0    1 -3.251e+03
     1    8 -3.251e+03  2.80e-05  5.41e-05  4.7e-05  8.0e+09  4.7e-06  2.16e+05
     2   16 -3.253e+03  3.53e-04  6.39e-04  2.9e-01  2.0e+00  4.1e-02  4.05e-01
     3   20 -3.263e+03  3.10e-03  9.22e-04  8.6e-01  6.0e-01  6.5e-01  3.78e-03
     4   28 -3.264e+03  3.85e-04  8.48e-04  3.7e-06  1.4e+01  5.2e-06  2.26e+00
     5   29 -3.264e+03  2.18e-06  1.92e-06  3.7e-06  2.0e+00  5.2e-06  4.51e+00
     6   30 -3.264e+03  1.28e-07  1.32e-07  3.7e-06  2.0e+00  5.2e-06  4.63e+00
     7   39 -3.269e+03  1.55e-03  1.37e-03  5.7e-02  2.0e+00  8.5e-02  4.62e+00
     8   41 -3.270e+03  3.03e-04  3.27e-04  1.1e-02  2.0e+00  1.7e-02  5.85e+01
     9   42 -3.271e+03  1.87e-04  3.03e-04  1.1e-02  2.0e+00  1.7e-02  3.44e+01
    10   43 -3.271e+03  7.07e-05  3.45e-04  1.0e-02  2.0e+00  1.7e-02  2.44e+00
    11   45 -3.271e+03  5.88e-05  1.70e-04  3.6e-03  2.0e+00  6.1e-03  1.59e-01
    12   46 -3.271e+03  4.26e-05  6.20e-05  3.6e-03  2.0e+00  6.1e-03  1.74e-02
    13   48 -3.271e+03  2.50e-06  3.71e-06  1.1e-03  2.0e+00  1.9e-03  3.68e-03
    14   54 -3.271e+03  4.76e-07  9.43e-07  5.1e-08  2.1e+01  8.5e-08  2.92e-03
    15   55 -3.271e+03  6.85e-10  1.07e-09  4.8e-08  2.0e+00  8.5e-08  1.85e-03
    16   65 -3.272e+03  1.16e-04  2.87e-04  1.6e-02  1.6e+00  2.9e-02  1.85e-03
    17   67 -3.274e+03  6.03e-04  6.38e-04  1.3e-02  6.0e-01  2.9e-02  2.63e-03
    18   68 -3.275e+03  3.50e-04  4.78e-04  1.4e-02  7.6e-01  2.9e-02  7.41e-04
    19   79 -3.275e+03  7.20e-06  1.72e-05  1.8e-07  2.8e+00  3.2e-07  2.32e-05
    20   80 -3.275e+03  1.31e-06  1.24e-06  1.5e-07  2.0e+00  3.2e-07  2.61e-06
    21   86 -3.275e+03  6.75e-07  7.48e-07  1.8e-04  8.3e-01  3.3e-04  1.33e-06
    22   87 -3.275e+03  2.72e-07  3.09e-07  1.6e-04  0.0e+00  3.9e-04  3.09e-07
    23   88 -3.275e+03  2.36e-09  6.67e-10  3.3e-05  0.0e+00  7.2e-05  6.67e-10
    24   99 -3.275e+03 -2.64e-15  1.40e-16  3.7e-15  9.9e+03  6.7e-15  1.35e-10

 ***** FALSE CONVERGENCE *****

 FUNCTION    -3.274708e+03   RELDX        3.716e-15
 FUNC. EVALS      99         GRAD. EVALS      24
 PRELDF       1.396e-16      NPRELDF      1.353e-10

     I      FINAL X(I)        D(I)          G(I)

     1    8.956549e-06     1.000e+00    -6.835e+01
     2    6.527503e-02     1.000e+00    -3.575e-02
     3    8.997967e-01     1.000e+00     1.025e-02


DASH-USD
                     Constant Mean - GARCH Model Results                      
==============================================================================
Dep. Variable:                   Diff   R-squared:                       0.000
Mean Model:             Constant Mean   Adj. R-squared:                  0.000
Vol Model:                      GARCH   Log-Likelihood:                1111.64
Distribution:                  Normal   AIC:                          -2215.28
Method:            Maximum Likelihood   BIC:                          -2198.96
                                        No. Observations:                  437
Date:                Wed, Dec 13 2023   Df Residuals:                      436
Time:                        15:54:33   Df Model:                            1
                                  Mean Model                                  
==============================================================================
                  coef    std err          t      P>|t|       95.0% Conf. Int.
------------------------------------------------------------------------------
mu         -6.4232e-04  8.411e-04     -0.764      0.445 [-2.291e-03,1.006e-03]
                              Volatility Model                              
============================================================================
                 coef    std err          t      P>|t|      95.0% Conf. Int.
----------------------------------------------------------------------------
omega      5.7952e-05  1.155e-05      5.017  5.237e-07 [3.531e-05,8.059e-05]
alpha[1]       0.1076  6.875e-02      1.565      0.117  [-2.712e-02,  0.242]
beta[1]        0.7379  7.224e-02     10.215  1.697e-24     [  0.596,  0.880]
============================================================================

Covariance estimator: robust

 ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 


     I     INITIAL X(I)        D(I)

     1     3.508099e-04     1.000e+00
     2     5.000000e-02     1.000e+00
     3     5.000000e-02     1.000e+00

    IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
     0    1 -1.499e+03
     1    7 -1.499e+03  7.94e-05  1.32e-04  1.0e-04  2.0e+09  1.0e-05  1.30e+05
     2    8 -1.499e+03  3.23e-06  3.54e-06  9.5e-05  2.0e+00  1.0e-05  8.66e-01
     3   15 -1.501e+03  9.78e-04  1.39e-03  2.9e-01  2.0e+00  4.1e-02  8.63e-01
     4   18 -1.505e+03  2.42e-03  2.42e-03  5.8e-01  1.9e+00  1.6e-01  2.04e-01
     5   19 -1.508e+03  2.14e-03  4.26e-03  4.2e-01  1.3e+00  3.3e-01  1.58e-02
     6   27 -1.508e+03  3.83e-04  8.03e-04  1.1e-05  5.3e+00  1.2e-05  2.13e-03
     7   28 -1.508e+03  3.48e-06  2.55e-06  9.5e-06  2.0e+00  1.2e-05  1.44e-04
     8   35 -1.508e+03  8.90e-05  1.11e-04  2.3e-02  5.2e-01  2.8e-02  1.34e-04
     9   37 -1.509e+03  1.68e-04  1.04e-04  2.8e-02  0.0e+00  3.7e-02  1.04e-04
    10   38 -1.509e+03  1.09e-04  1.02e-04  5.8e-02  0.0e+00  7.1e-02  1.02e-04
    11   39 -1.509e+03  2.89e-06  6.47e-06  4.4e-03  0.0e+00  6.7e-03  6.47e-06
    12   40 -1.509e+03  5.70e-07  4.53e-07  6.0e-04  0.0e+00  7.4e-04  4.53e-07
    13   41 -1.509e+03  7.55e-08  3.13e-09  2.3e-04  0.0e+00  3.2e-04  3.13e-09
    14   42 -1.509e+03 -4.18e-09  4.77e-11  2.0e-05  0.0e+00  3.0e-05  4.77e-11

 ***** RELATIVE FUNCTION CONVERGENCE *****

 FUNCTION    -1.508897e+03   RELDX        1.964e-05
 FUNC. EVALS      42         GRAD. EVALS      14
 PRELDF       4.769e-11      NPRELDF      4.769e-11

     I      FINAL X(I)        D(I)          G(I)

     1    9.604329e-05     1.000e+00     1.344e+01
     2    1.505650e-01     1.000e+00    -3.849e-03
     3    6.041880e-01     1.000e+00     5.082e-03


XRP-USD
                     Constant Mean - GARCH Model Results                      
==============================================================================
Dep. Variable:                   Diff   R-squared:                       0.000
Mean Model:             Constant Mean   Adj. R-squared:                  0.000
Vol Model:                      GARCH   Log-Likelihood:                2047.80
Distribution:                  Normal   AIC:                          -4087.60
Method:            Maximum Likelihood   BIC:                          -4068.65
                                        No. Observations:                  842
Date:                Wed, Dec 13 2023   Df Residuals:                      841
Time:                        15:54:33   Df Model:                            1
                                  Mean Model                                  
==============================================================================
                  coef    std err          t      P>|t|       95.0% Conf. Int.
------------------------------------------------------------------------------
mu         -4.6731e-04  6.177e-04     -0.757      0.449 [-1.678e-03,7.433e-04]
                              Volatility Model                              
============================================================================
                 coef    std err          t      P>|t|      95.0% Conf. Int.
----------------------------------------------------------------------------
omega      1.5062e-05  1.386e-11  1.087e+06      0.000 [1.506e-05,1.506e-05]
alpha[1]       0.1002  4.460e-02      2.247  2.463e-02   [1.281e-02,  0.188]
beta[1]        0.8785  3.225e-02     27.244 1.942e-163     [  0.815,  0.942]
============================================================================

Covariance estimator: robust

 ***** ESTIMATION WITH ANALYTICAL GRADIENT ***** 


     I     INITIAL X(I)        D(I)

     1     5.269160e-04     1.000e+00
     2     5.000000e-02     1.000e+00
     3     5.000000e-02     1.000e+00

    IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
     0    1 -2.727e+03
     1    7 -2.728e+03  2.94e-04  4.98e-04  3.0e-04  1.5e+09  3.0e-05  3.84e+05
     2    8 -2.728e+03  8.05e-06  8.84e-06  2.9e-04  2.0e+00  3.0e-05  1.09e+01
     3   15 -2.747e+03  7.08e-03  1.08e-02  5.5e-01  2.0e+00  1.2e-01  1.07e+01
     4   17 -2.763e+03  5.63e-03  5.08e-03  3.0e-01  2.0e+00  1.2e-01  2.49e+00
     5   19 -2.795e+03  1.15e-02  1.13e-02  3.9e-01  2.0e+00  2.4e-01  2.86e+02
     6   21 -2.801e+03  2.26e-03  2.28e-03  5.3e-02  2.0e+00  4.9e-02  5.25e+01
     7   23 -2.817e+03  5.47e-03  9.00e-03  1.8e-01  2.0e+00  2.2e-01  7.28e-01
     8   30 -2.818e+03  3.64e-04  8.98e-04  5.8e-06  1.4e+01  7.7e-06  3.52e-02
     9   31 -2.818e+03  8.77e-06  7.09e-06  5.4e-06  2.0e+00  7.7e-06  4.19e-03

     I     INITIAL X(I)        D(I)

     1     5.269160e-04     1.000e+00
     2     5.000000e-02     1.000e+00
     3     5.000000e-02     1.000e+00

    IT   NF      F         RELDF    PRELDF    RELDX   STPPAR   D*STEP   NPRELDF
     0    1 -2.727e+03
     1    7 -2.728e+03  2.94e-04  4.98e-04  3.0e-04  1.5e+09  3.0e-05  3.84e+05
     2    8 -2.728e+03  8.05e-06  8.84e-06  2.9e-04  2.0e+00  3.0e-05  1.09e+01
     3   15 -2.747e+03  7.08e-03  1.08e-02  5.5e-01  2.0e+00  1.2e-01  1.07e+01
     4   17 -2.763e+03  5.63e-03  5.08e-03  3.0e-01  2.0e+00  1.2e-01  2.49e+00
     5   19 -2.795e+03  1.15e-02  1.13e-02  3.9e-01  2.0e+00  2.4e-01  2.86e+02
     6   21 -2.801e+03  2.26e-03  2.28e-03  5.3e-02  2.0e+00  4.9e-02  5.25e+01
     7   23 -2.817e+03  5.47e-03  9.00e-03  1.8e-01  2.0e+00  2.2e-01  7.28e-01
     8   30 -2.818e+03  3.64e-04  8.98e-04  5.8e-06  1.4e+01  7.7e-06  3.52e-02
     9   31 -2.818e+03  8.77e-06  7.09e-06  5.4e-06  2.0e+00  7.7e-06  4.19e-03
    10   32 -2.818e+03  4.47e-07  3.78e-07  5.4e-06  2.0e+00  7.7e-06  3.84e-03
    11   39 -2.818e+03  9.72e-05  1.83e-04  1.1e-02  1.9e+00  1.6e-02  3.85e-03
    12   41 -2.818e+03  1.72e-04  8.90e-05  1.3e-02  0.0e+00  2.0e-02  8.90e-05
    13   43 -2.820e+03  5.45e-04  5.33e-04  4.7e-02  0.0e+00  7.8e-02  9.20e-04
    14   44 -2.821e+03  2.62e-04  3.21e-04  4.1e-02  6.2e-01  7.8e-02  4.25e-04
    15   45 -2.821e+03  2.78e-05  5.41e-05  1.2e-02  0.0e+00  2.2e-02  5.41e-05
    16   46 -2.821e+03  1.08e-05  1.22e-05  2.1e-03  0.0e+00  3.2e-03  1.22e-05
    17   47 -2.821e+03  7.41e-08  1.61e-07  7.0e-04  0.0e+00  1.2e-03  1.61e-07
    18   48 -2.821e+03  2.95e-08  8.37e-09  2.4e-04  0.0e+00  4.7e-04  8.37e-09
    19   49 -2.821e+03  1.95e-09  1.55e-10  1.6e-05  0.0e+00  2.9e-05  1.55e-10
    20   50 -2.821e+03 -2.17e-09  1.69e-11  6.0e-06  0.0e+00  1.0e-05  1.69e-11

 ***** RELATIVE FUNCTION CONVERGENCE *****

 FUNCTION    -2.820737e+03   RELDX        6.033e-06
 FUNC. EVALS      50         GRAD. EVALS      20
 PRELDF       1.693e-11      NPRELDF      1.693e-11

     I      FINAL X(I)        D(I)          G(I)

     1    2.894236e-05     1.000e+00    -5.379e+01
     2    2.073207e-01     1.000e+00    -1.227e-03
     3    7.706760e-01     1.000e+00    -6.875e-03

Per valutare la bontà del modello procediamo con un analisi sui residui.

Valutiamo:

  1. Stazionarietà
  2. Autocorrelazione
  3. Eteroschedasticità
In [ ]:
# STAZIONARIETÀ
kpss_test(residualsDict, "Resid", cryptoList)
adf_test(residualsDict, "Resid", cryptoList)

# AUTOCORRELAZIONE
ljung_box_test(residualsDict, "Resid", cryptoList)

# ETEROSCHEDASTICITÀ
breusch_pagan_test(residualsDict, "Resid", cryptoList)
white_test(residualsDict, "Resid", cryptoList)

                        KPSS                         
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃ pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 0.17432400371354975 │    0.1 │ Accept  │
│ BTC-USD  │ 0.19844539579445133 │    0.1 │ Accept  │
│ DASH-USD │ 0.11788575359356059 │    0.1 │ Accept  │
│ XRP-USD  │ 0.10803807541946762 │    0.1 │ Accept  │
└──────────┴─────────────────────┴────────┴─────────┘

                                 ADF                                 
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃                 pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ -15.693567424856559 │ 1.4241061163949666e-28 │ Reject  │
│ BTC-USD  │ -29.079199700789154 │                    0.0 │ Reject  │
│ DASH-USD │ -20.551692589739947 │                    0.0 │ Reject  │
│ XRP-USD  │ -30.272866468250122 │                    0.0 │ Reject  │
└──────────┴─────────────────────┴────────────────────────┴─────────┘

                            Ljung-Box                            
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃              pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 12.429167305858956 │  0.2573598822424017 │ Accept  │
│ BTC-USD  │ 11.049385706036809 │ 0.35368336960583363 │ Accept  │
│ DASH-USD │  7.738434333520586 │  0.6543690599625923 │ Accept  │
│ XRP-USD  │  13.16396100965736 │ 0.21464853827574287 │ Accept  │
└──────────┴────────────────────┴─────────────────────┴─────────┘

                           Breusch-Pagan                           
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃           StatValue ┃               pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │  3.0552372793185714 │   0.0808291850355068 │ Accept  │
│ BTC-USD  │    4.81037750869296 │ 0.028547494279450508 │ Reject  │
│ DASH-USD │ 0.16268201483311248 │   0.6868977927761047 │ Accept  │
│ XRP-USD  │   6.365618133810369 │ 0.011819036032203746 │ Reject  │
└──────────┴─────────────────────┴──────────────────────┴─────────┘

                              White                               
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━┓
┃ Crypto   ┃          StatValue ┃               pValue ┃ Outcome ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━┩
│ ETH-USD  │ 2.1579503833990525 │  0.11617918995361964 │ Accept  │
│ BTC-USD  │ 2.5589215991266947 │  0.07795936137356213 │ Accept  │
│ DASH-USD │  1.432578207289038 │   0.2398217511979006 │ Accept  │
│ XRP-USD  │ 3.4750426643481562 │ 0.031407409545298644 │ Reject  │
└──────────┴────────────────────┴──────────────────────┴─────────┘

Grafichiamo quindi i residui ottenuti.

In [ ]:
plt.figure(figsize= (16,9))
fig, axs = plt.subplots(nrows=len(cryptoList), figsize=(20, 15))
i = 0
for crypto in cryptoList:
    axs[i].scatter(x = np.arange(0, residualsDict[crypto].shape[0]), y = residualsDict[crypto]["Resid"], sizes = [5])
    axs[i].set_title("GARCH residuals for " + crypto)
    i += 1


plt.show()
plt.clf()

<Figure size 1600x900 with 0 Axes>
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

Per quanto riguarda l'autocorrelazione andiamo a produrre degli autocorrelogrammi.

In [ ]:
correlograms(
    residualsDict, 
    "Resid", 
    cryptoList,
    titleAuto = "Restricted Differencing (Quadratic) Autocorrelation",
    titlePartial = "Restricted Differencing (Quadratic) Partial-Autocorrelation",
    zeroAuto = False, zeroPar = False,
    yLimsAuto = True, yLimsPar = True
)
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

6.1. Approccio non parametrico (Tecnica di Bootstrap)¶

Con l'obiettivo di generare delle bande di predizione in cui ci aspettiamo che ricada il nostro testing set andiamo ad applicare la tecnica di Bootstrap. Questa tecnica, che sembra empiricamente funzionare discretamente bene, è molto facile e veloce rispetto ad un tradizionale approccio parametrico ma non dà informazioni complete sul processo generatore della serie dei residui.

Procediamo quindi con i seguenti passi per ogni cryptovaluta:

  1. Fit del modello GARCH(1,1) sul testingSet dei rendimenti
  2. Estrazione dei parametri del modello ($\alpha_0$, $\alpha_1$, $\beta_1$)
  3. Tecnica di Bootstrap:
    • Estrazione di TESTING_SET_LEN samples con rimpiazzamento dai residui del modello
    • Generazione di 10000 traiettorie:
      $\sigma^2_t = \alpha_0 + \alpha_1 X_{t-1}^2 + \beta_1 \sigma_{t-1}^2$
      $X_t = \sigma_t W_t$
      $\forall t = 1,...,testSetLen$
      Dove $\sigma_0, X_0 = lastTrainingSetPoint$
In [ ]:
NUM_SAMPLES = 10000
fig, axs = plt.subplots(nrows = len(cryptoList), ncols = 1, figsize=(20,15))
axsIndex = 0
np.random.seed(NP_RANDOM_SEED)
trajectoriesDict : dict[str, list[list]] = {}

r['source']('./R/R_Scripts.R')
garchResiduals_R = robjects.globalenv['garchResiduals']

for crypto in cryptoList :
    TRAINING_SET_LEN = trainCryptoDiffDict[crypto]["Diff"].shape[0]
    TESTING_SET_LEN = testCryptoDiffDict[crypto]["Diff"].shape[0]
    garchModel = arch_model(trainCryptoDiffDict[crypto]["Diff"])
    fittedModel = garchModel.fit(disp = 'off')
    residuals = residualsDict[crypto]["Resid"]

    alpha_0 = fittedModel.params["omega"]
    alpha_1 = fittedModel.params["alpha[1]"]
    beta_1 = fittedModel.params["beta[1]"]

    sigma_0 = trainCryptoDiffDict[crypto]["Diff"][TRAINING_SET_LEN - 1]
    trajectories = []
    for _ in range(NUM_SAMPLES):
        computedXt = []
        bootstrapSample = np.random.choice(a = residuals, size = TESTING_SET_LEN, replace = True)
        prev_Xt = trainCryptoDiffDict[crypto]["Diff"].iloc[TRAINING_SET_LEN-1]
        computedXt.append(prev_Xt)
        prev_var_t = sigma_0**2
        for i in range(TESTING_SET_LEN):
            var_t = alpha_0 + (alpha_1 * (prev_Xt**2)) + (beta_1 * prev_var_t)
            Xt = np.sqrt(var_t) * bootstrapSample[i]
            computedXt.append(Xt)
            prev_Xt = Xt
            prev_var_t = var_t
        trajectories.append(computedXt)
    trajectoriesDict[crypto] = trajectories

    axs[axsIndex].set_title(crypto + " Bootstrap Generated trajectories")
    for traj in trajectories:
        axs[axsIndex].plot(traj)
    axsIndex += 1

plt.show()
plt.clf()
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

Partendo dalle traiettorie generate, andiamo a calcolare i quantili empirici per ogni t e costruiamo le nostre bande di predizione al 10% e 90%.

Grafichiamo poi il tutto sui dati di testing effettivi.

In [ ]:
LOW_PERC = 0.10
HIGH_PERC = 0.90
In [ ]:
fig, axs = plt.subplots(nrows = len(cryptoList), ncols = 1, figsize=(20,15))
axsIndex = 0
trajQuantDict : dict[str,tuple[list,list]] = {}

for crypto in cryptoList:
    TRAINING_SET_LEN = trainCryptoDiffDict[crypto].shape[0]
    TESTING_SET_LEN = testCryptoDiffDict[crypto].shape[0]
    
    trajectories = trajectoriesDict[crypto]
    trajectories = np.array(trajectories)
    trajectories = trajectories.T
    trajQuantList_25 = []
    trajQuantList_75 = []
    for i in range(trajectories.shape[0]):
        trajQuant_25 = np.quantile(a = trajectories[i], q = LOW_PERC)
        trajQuant_75 = np.quantile(a = trajectories[i], q = HIGH_PERC)
        trajQuantList_25.append(trajQuant_25)
        trajQuantList_75.append(trajQuant_75)
    trajQuantDict[crypto] = (trajQuantList_25,trajQuantList_75)

    firstPoint = trainCryptoDiffDict[crypto]["Diff"].iloc[TRAINING_SET_LEN - 1]
    axs[axsIndex].plot(np.arange(1, TESTING_SET_LEN + 1, 1), testCryptoDiffDict[crypto]["Diff"], color = colors[axsIndex])
    axs[axsIndex].scatter(x = 0, y = firstPoint, color = colors[axsIndex], sizes = [10])
    axs[axsIndex].fill_between(x = np.arange(0, len(trajQuantList_25), 1), y1 = trajQuantList_25, y2 = trajQuantList_75, alpha = 0.20, color = colors[axsIndex])
    axs[axsIndex].set_title(crypto + " Bootstrap Prediction Intervals")

    axsIndex += 1

plt.show()
plt.clf()
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>
In [ ]:
inIntervalTable = buildTable("TestingSet captured by interval - Non Parametric Approch", ["Crypto", "% of Points"], ["left", "right"])

for crypto in cryptoList:
    inInterval = 0
    lowTraj = trajQuantDict[crypto][0][1:]
    highTraj = trajQuantDict[crypto][1][1:]
    TESTING_SET_LEN = testCryptoDiffDict[crypto].shape[0]
    for t in range(0, TESTING_SET_LEN):
        if lowTraj[t] <= testCryptoDiffDict[crypto]["Diff"][t] <= highTraj[t]:
            inInterval+=1
    inIntervalTable.add_row(crypto, str((inInterval/TESTING_SET_LEN) * 100))

console = Console()
console.print(inIntervalTable)

TestingSet captured by interval 
    - Non Parametric Approch    
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┓
┃ Crypto   ┃       % of Points ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━┩
│ ETH-USD  │ 88.65979381443299 │
│ BTC-USD  │ 83.83838383838383 │
│ DASH-USD │ 89.58333333333334 │
│ XRP-USD  │ 83.87096774193549 │
└──────────┴───────────────────┘

6.2. Approccio Parametrico¶

Per l'approccio parametrico abbiamo 2 guess sulla distribuzione dei residui:

  • Skewed Student
  • Generalized Logistic

Andiamo quindi a vedere quale tra le due, per ogni crypto, cattura meglio i residui del modello GARCH.

In [ ]:
np.random.seed(seed=NP_RANDOM_SEED)

figure, axs = plt.subplots(nrows = 4, ncols = 2, figsize = (15, 25))
skewedStudentparamsDict : dict[str, tuple[float, float, float, float]] = {}
genLogisticParamsDict : dict[str, tuple[float, float, float, float]] = {}

axsIndex = 0
for crypto in cryptoList :
    RESIDUALS_LEN = residualsDict[crypto]["Resid"].shape[0]
    fitResults = fit(nct, residualsDict[crypto]["Resid"], bounds=[(0, 30), (0, 30), (-30, 30), (0, 30)])
    skewedStudentparamsDict[crypto] = (fitResults.params[0], fitResults.params[1], fitResults.params[2], fitResults.params[3], fitResults.nllf())
    
    fitResults.plot(ax = axs[axsIndex][0])
    axs[axsIndex][0].set_title("Data Histogram VS Skewed Student\nfor " + crypto)
    axs[axsIndex][0].set_xlabel("")
    axs[axsIndex][0].set_ylabel("")
    axs[axsIndex][0].get_legend().remove()
    axs[axsIndex][0].properties()['children'][0].set_color("black")   
    for elem in axs[axsIndex][0].properties()['children'][1:-2] :
        if type(elem) == matplotlib.patches.Rectangle:
            elem.set_color(colors[axsIndex])      
    axsIndex += 1

axsIndex = 0
for crypto in cryptoList :
    RESIDUALS_LEN = residualsDict[crypto]["Resid"].shape[0]
    fitResults = fit(genlogistic, residualsDict[crypto]["Resid"], bounds=[(0, 30), (-30, 30), (0, 30)])
    genLogisticParamsDict[crypto] = (fitResults.params[0], fitResults.params[1], fitResults.params[2] , fitResults.nllf())

    fitResults.plot(ax = axs[axsIndex][1])
    axs[axsIndex][1].set_title("Data Histogram VS Generalized Logistic\nfor " + crypto)
    axs[axsIndex][1].set_xlabel("")
    axs[axsIndex][1].set_ylabel("")
    axs[axsIndex][1].get_legend().remove()
    axs[axsIndex][1].properties()['children'][0].set_color("black")   
    for elem in axs[axsIndex][1].properties()['children'][1:-2] : 
        if type(elem) == matplotlib.patches.Rectangle:
            elem.set_color(colors[axsIndex])   
    axsIndex += 1

paramsSkewedTable = buildTable("Skewed Student Params", ["Crypto", "Degrees Of Freedom", "Non Centrality Param", "Mean", "Scale", "Negative Loglikelihood"], ["left", "right", "right",  "right", "right", "right"])
for crypto in cryptoList :
    paramsSkewedTable.add_row(crypto, str(skewedStudentparamsDict[crypto][0]), str(skewedStudentparamsDict[crypto][1]), str(skewedStudentparamsDict[crypto][2]), str(skewedStudentparamsDict[crypto][3]), str(skewedStudentparamsDict[crypto][4]))

paramsLogisticTable = buildTable("Generalized Logistic Params", ["Crypto", "C Param", "Mean", "Scale", "Negative LogLikelihood"], ["left", "right",  "right", "right", "right"])
for crypto in cryptoList :
    paramsLogisticTable.add_row(crypto, str(genLogisticParamsDict[crypto][0]), str(genLogisticParamsDict[crypto][1]), str(genLogisticParamsDict[crypto][2]), str(genLogisticParamsDict[crypto][3]))

console = Console()
console.print(paramsSkewedTable)
console.print(paramsLogisticTable)

plt.show()
plt.clf()

                                               Skewed Student Params                                               
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┓
┃          ┃                    ┃     Non Centrality ┃                   ┃                    ┃          Negative ┃
┃ Crypto   ┃ Degrees Of Freedom ┃              Param ┃              Mean ┃              Scale ┃     Loglikelihood ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━┩
│ ETH-USD  │  3.516473494404673 │ 0.033722144524455… │ 0.02062388621608… │ 0.7193371332955668 │ 1207.03475733531… │
│ BTC-USD  │ 3.3064687645793187 │ 0.009557726120843… │ 0.00147848072745… │ 0.6790429283134127 │ 1203.49576520300… │
│ DASH-USD │  3.824098827970277 │                0.0 │ 0.01404492392671… │  0.709022690041867 │  588.797628081744 │
│ XRP-USD  │ 3.3918528112268227 │                0.0 │ -0.0046996569766… │  0.658324631240604 │ 1104.11340489189… │
└──────────┴────────────────────┴────────────────────┴───────────────────┴────────────────────┴───────────────────┘

                                     Generalized Logistic Params                                     
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Crypto   ┃            C Param ┃                Mean ┃              Scale ┃ Negative LogLikelihood ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━┩
│ ETH-USD  │ 0.7637779420519504 │ 0.23762413558863482 │ 0.4807130762122873 │     1205.3666340245725 │
│ BTC-USD  │ 0.8578133191926725 │ 0.11784578310968241 │ 0.4963510618643525 │     1214.5304615377536 │
│ DASH-USD │ 0.5826316332545343 │ 0.38210941598910697 │ 0.4148984939135314 │      587.7888993819361 │
│ XRP-USD  │ 0.8396765179300889 │ 0.11485328042880677 │ 0.4703879989799512 │     1121.7298020281225 │
└──────────┴────────────────────┴─────────────────────┴────────────────────┴────────────────────────┘
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>

Per decidere la distribuzione più adatta ricordiamo che la minimizzazione della Negative Log-Likelihood è un problema proxy per il problema di massimizzazione della Maximum Likelihood". Procediamo quindi a selezionare la distribuzione che restituisce il valore minimo di Negative Log-Likelihood".

In [ ]:
bestDistrTable = buildTable("Best Guessed Distribution for crypto", ["Crypto", "Student Neg Log-Likelihood", "Logistic Neg Log-Likelihood","Chosen Distribution"], ["left", "right", "right", "right"])
for crypto in cryptoList:
    if genLogisticParamsDict[crypto][3] > skewedStudentparamsDict[crypto][4]:
        bestDistrTable.add_row(crypto, str(skewedStudentparamsDict[crypto][4]), str(genLogisticParamsDict[crypto][3]), "Skewed Student")
    else:
        bestDistrTable.add_row(crypto, str(skewedStudentparamsDict[crypto][4]), str(genLogisticParamsDict[crypto][3]), "Generalized Logistic")
Console().print(bestDistrTable)

                             Best Guessed Distribution for crypto                             
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━┓
┃ Crypto   ┃ Student Neg Log-Likelihood ┃ Logistic Neg Log-Likelihood ┃  Chosen Distribution ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━┩
│ ETH-USD  │         1207.0347573353117 │          1205.3666340245725 │ Generalized Logistic │
│ BTC-USD  │         1203.4957652030082 │          1214.5304615377536 │       Skewed Student │
│ DASH-USD │           588.797628081744 │           587.7888993819361 │ Generalized Logistic │
│ XRP-USD  │         1104.1134048918952 │          1121.7298020281225 │       Skewed Student │
└──────────┴────────────────────────────┴─────────────────────────────┴──────────────────────┘
In [ ]:
NUM_SAMPLES = 10000
fig, axs = plt.subplots(nrows = len(cryptoList), ncols = 1, figsize=(20,15))
axsIndex = 0
np.random.seed(NP_RANDOM_SEED)
paramTrajectoriesDict : dict[str, list[list]] = {}


for crypto in cryptoList:
    TRAINING_SET_LEN = trainCryptoDiffDict[crypto].shape[0]
    TESTING_SET_LEN = testCryptoDiffDict[crypto].shape[0]
    trajectories = []
    for _ in range(NUM_SAMPLES):
        computedXt = []
        if genLogisticParamsDict[crypto][3] > skewedStudentparamsDict[crypto][4]:
            degreesOfFreedom = skewedStudentparamsDict[crypto][0]
            nonCentralityParam = skewedStudentparamsDict[crypto][1]
            loc = skewedStudentparamsDict[crypto][2]
            scale = skewedStudentparamsDict[crypto][3]
            residuals = nct.rvs(degreesOfFreedom, nonCentralityParam, size = TESTING_SET_LEN, loc = loc, scale = scale)
        else:
            cParam = genLogisticParamsDict[crypto][0]
            loc = genLogisticParamsDict[crypto][1]
            scale = genLogisticParamsDict[crypto][2]
            residuals = genlogistic.rvs(cParam, size = TESTING_SET_LEN, loc = loc, scale = scale)
        prev_Xt = trainCryptoDiffDict[crypto]["Diff"].iloc[TRAINING_SET_LEN-1]
        computedXt.append(prev_Xt)
        prev_var_t = sigma_0**2
        for i in range(TESTING_SET_LEN):
            var_t = alpha_0 + (alpha_1 * (prev_Xt**2)) + (beta_1 * prev_var_t)
            Xt = np.sqrt(var_t) * residuals[i]
            computedXt.append(Xt)
            prev_Xt = Xt
            prev_var_t = var_t
        trajectories.append(computedXt)
    paramTrajectoriesDict[crypto] = trajectories

    axs[axsIndex].set_title(crypto + " Parametric Approch Generated trajectories")
    for traj in trajectories:
        axs[axsIndex].plot(traj)
    axsIndex += 1

plt.show()
plt.clf()
No description has been provided for this image 
<Figure size 640x480 with 0 Axes>
In [ ]:
fig, axs = plt.subplots(nrows = len(cryptoList), ncols = 1, figsize=(20,15))
axsIndex = 0
paramTrajQuantDict : dict[str,tuple[list,list]] = {}

for crypto in cryptoList:
    TRAINING_SET_LEN = trainCryptoDiffDict[crypto].shape[0]
    TESTING_SET_LEN = testCryptoDiffDict[crypto].shape[0]
    
    trajectories = paramTrajectoriesDict[crypto]
    trajectories = np.array(trajectories)
    trajectories = trajectories.T
    trajQuantList_25 = []
    trajQuantList_75 = []
    for i in range(trajectories.shape[0]):
        trajQuant_25 = np.quantile(a = trajectories[i], q = LOW_PERC)
        trajQuant_75 = np.quantile(a = trajectories[i], q = HIGH_PERC)
        trajQuantList_25.append(trajQuant_25)
        trajQuantList_75.append(trajQuant_75)
    paramTrajQuantDict[crypto] = (trajQuantList_25,trajQuantList_75)

    firstPoint = trainCryptoDiffDict[crypto]["Diff"].iloc[TRAINING_SET_LEN - 1]
    axs[axsIndex].plot(np.arange(1, TESTING_SET_LEN + 1, 1), testCryptoDiffDict[crypto]["Diff"], color = colors[axsIndex])
    axs[axsIndex].scatter(x = 0, y = firstPoint, color = colors[axsIndex], sizes = [10])
    axs[axsIndex].fill_between(x = np.arange(0, len(trajQuantList_25), 1), y1 = trajQuantList_25, y2 = trajQuantList_75, alpha = 0.20, color = colors[axsIndex])
    axs[axsIndex].set_title(crypto + " Bootstrap Prediction Intervals")

    axsIndex += 1

plt.show()
plt.clf()
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<Figure size 640x480 with 0 Axes>
In [ ]:
inIntervalTable = buildTable("TestingSet captured by interval - Parametric Approch", ["Crypto", "% of Points"], ["left", "right"])

for crypto in cryptoList:
    inInterval = 0
    lowTraj = paramTrajQuantDict[crypto][0][1:]
    highTraj = paramTrajQuantDict[crypto][1][1:]
    TESTING_SET_LEN = testCryptoDiffDict[crypto].shape[0]
    for t in range(0, TESTING_SET_LEN):
        if lowTraj[t] <= testCryptoDiffDict[crypto]["Diff"][t] <= highTraj[t]:
            inInterval+=1
    inIntervalTable.add_row(crypto, str((inInterval/TESTING_SET_LEN) * 100))

console = Console()
console.print(inIntervalTable)

TestingSet captured by interval 
      - Parametric Approch      
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┓
┃ Crypto   ┃       % of Points ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━┩
│ ETH-USD  │ 91.75257731958763 │
│ BTC-USD  │ 92.92929292929293 │
│ DASH-USD │ 85.41666666666666 │
│ XRP-USD  │ 84.94623655913979 │
└──────────┴───────────────────┘
In [ ]:
inIntervalTable = buildTable("TestingSet captured by interval - Comparison", ["Crypto", "% of Points - Param", "% of Points - Non Param"], ["left", "right", "right"])

for crypto in cryptoList:
    TESTING_SET_LEN = testCryptoDiffDict[crypto].shape[0]
    paramInInterval = 0
    nonParamInInterval = 0
    lowTraj = paramTrajQuantDict[crypto][0][1:]
    highTraj = paramTrajQuantDict[crypto][1][1:]
    
    for t in range(0, TESTING_SET_LEN):
        if lowTraj[t] <= testCryptoDiffDict[crypto]["Diff"][t] <= highTraj[t]:
            paramInInterval+=1
    
    lowTraj = trajQuantDict[crypto][0][1:]
    highTraj = trajQuantDict[crypto][1][1:]
    for t in range(0, TESTING_SET_LEN):
        if lowTraj[t] <= testCryptoDiffDict[crypto]["Diff"][t] <= highTraj[t]:
            nonParamInInterval+=1
    inIntervalTable.add_row(crypto, str((paramInInterval/TESTING_SET_LEN) * 100), str((nonParamInInterval/TESTING_SET_LEN) * 100))

console = Console()
console.print(inIntervalTable)

        TestingSet captured by interval - Comparison        
┏━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━━━━━━━┓
┃ Crypto   ┃ % of Points - Param ┃ % of Points - Non Param ┃
┡━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━━━━━━━┩
│ ETH-USD  │   91.75257731958763 │       88.65979381443299 │
│ BTC-USD  │   92.92929292929293 │       83.83838383838383 │
│ DASH-USD │   85.41666666666666 │       89.58333333333334 │
│ XRP-USD  │   84.94623655913979 │       83.87096774193549 │
└──────────┴─────────────────────┴─────────────────────────┘